Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - basemath - lfun.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.16.1 lcov report (development 28904-c3aa21e911) Lines: 1477 1513 97.6 %
Date: 2023-12-04 07:51:13 Functions: 158 159 99.4 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2015  The PARI group.
       2             : 
       3             : This file is part of the PARI/GP package.
       4             : 
       5             : PARI/GP is free software; you can redistribute it and/or modify it under the
       6             : terms of the GNU General Public License as published by the Free Software
       7             : Foundation; either version 2 of the License, or (at your option) any later
       8             : version. It is distributed in the hope that it will be useful, but WITHOUT
       9             : ANY WARRANTY WHATSOEVER.
      10             : 
      11             : Check the License for details. You should have received a copy of it, along
      12             : with the package; see the file 'COPYING'. If not, write to the Free Software
      13             : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
      14             : 
      15             : /********************************************************************/
      16             : /**                                                                **/
      17             : /**                       L-functions                              **/
      18             : /**                                                                **/
      19             : /********************************************************************/
      20             : 
      21             : #include "pari.h"
      22             : #include "paripriv.h"
      23             : 
      24             : #define DEBUGLEVEL DEBUGLEVEL_lfun
      25             : 
      26             : /*******************************************************************/
      27             : /*  Accessors                                                      */
      28             : /*******************************************************************/
      29             : 
      30             : static GEN
      31       11749 : mysercoeff(GEN x, long n)
      32             : {
      33       11749 :   long N = n - valser(x);
      34       11749 :   return (N < 0)? gen_0: gel(x, N+2);
      35             : }
      36             : 
      37             : long
      38       12733 : ldata_get_type(GEN ldata) { return mael3(ldata, 1, 1, 1); }
      39             : 
      40             : GEN
      41       42021 : ldata_get_an(GEN ldata) { return gel(ldata, 1); }
      42             : 
      43             : GEN
      44       35413 : ldata_get_dual(GEN ldata) { return gel(ldata, 2); }
      45             : 
      46             : long
      47        2280 : ldata_isreal(GEN ldata) { return isintzero(gel(ldata, 2)); }
      48             : 
      49             : GEN
      50      238344 : ldata_get_gammavec(GEN ldata) { return gel(ldata, 3); }
      51             : 
      52             : long
      53       13809 : ldata_get_degree(GEN ldata) { return lg(gel(ldata, 3))-1; }
      54             : 
      55             : GEN
      56       99421 : ldata_get_k(GEN ldata)
      57             : {
      58       99421 :   GEN w = gel(ldata,4);
      59       99421 :   if (typ(w) == t_VEC) w = gel(w,1);
      60       99421 :   return w;
      61             : }
      62             : 
      63             : /* a_n = O(n^{k1 + epsilon}) */
      64             : GEN
      65          98 : ldata_get_k1(GEN ldata)
      66             : {
      67          98 :   GEN w = gel(ldata,4);
      68          98 :   if (typ(w) == t_VEC) return gel(w,2);
      69             :   /* by default, assume that k1 = k-1 and even (k-1)/2 for entire functions */
      70          98 :   w = gaddgs(w,-1);
      71          98 :   return ldata_get_residue(ldata)? w: gmul2n(w, -1);
      72             : }
      73             : 
      74             : /* a_n = O(n^{k1 + epsilon}) */
      75             : static double
      76       74464 : ldata_get_k1_dbl(GEN ldata)
      77             : {
      78       74464 :   GEN w = gel(ldata,4);
      79             :   double k;
      80       74464 :   if (typ(w) == t_VEC) return gtodouble(gel(w,2));
      81             :   /* by default, assume that k1 = k-1 and even (k-1)/2 for entire functions */
      82       73043 :   k = gtodouble(w);
      83       73043 :   return ldata_get_residue(ldata)? k-1: (k-1)/2.;
      84             : }
      85             : 
      86             : GEN
      87      170651 : ldata_get_conductor(GEN ldata) { return gel(ldata, 5); }
      88             : 
      89             : GEN
      90       68164 : ldata_get_rootno(GEN ldata) { return gel(ldata, 6); }
      91             : 
      92             : GEN
      93      128549 : ldata_get_residue(GEN ldata) { return lg(ldata) == 7 ? NULL: gel(ldata, 7); }
      94             : 
      95             : long
      96       93074 : linit_get_type(GEN linit) { return mael(linit, 1, 1); }
      97             : 
      98             : GEN
      99      139330 : linit_get_ldata(GEN linit) { return gel(linit, 2); }
     100             : 
     101             : GEN
     102      175415 : linit_get_tech(GEN linit) { return gel(linit, 3); }
     103             : 
     104             : long
     105      167565 : is_linit(GEN data)
     106             : {
     107      112170 :   return lg(data) == 4 && typ(data) == t_VEC
     108      279735 :                        && typ(gel(data, 1)) == t_VECSMALL;
     109             : }
     110             : 
     111             : GEN
     112       21598 : lfun_get_step(GEN tech) { return gmael(tech, 2, 1);}
     113             : 
     114             : GEN
     115       21598 : lfun_get_pol(GEN tech) { return gmael(tech, 2, 2);}
     116             : 
     117             : GEN
     118        5066 : lfun_get_Residue(GEN tech) { return gmael(tech, 2, 3);}
     119             : 
     120             : GEN
     121       34329 : lfun_get_k2(GEN tech) { return gmael(tech, 3, 1);}
     122             : 
     123             : GEN
     124       13130 : lfun_get_w2(GEN tech) { return gmael(tech, 3, 2);}
     125             : 
     126             : GEN
     127       13130 : lfun_get_expot(GEN tech) { return gmael(tech, 3, 3);}
     128             : 
     129             : GEN
     130        5033 : lfun_get_factgammavec(GEN tech) { return gmael(tech, 3, 4); }
     131             : 
     132             : /* Handle complex Vga whose sum is real */
     133             : static GEN
     134       80946 : sumVga(GEN Vga) { return real_i(vecsum(Vga)); }
     135             : /* sum_i max (Im v[i],0) */
     136             : static double
     137       17768 : sumVgaimpos(GEN v)
     138             : {
     139       17768 :   double d = 0.;
     140       17768 :   long i, l = lg(v);
     141       47557 :   for (i = 1; i < l; i++)
     142             :   {
     143       29789 :     GEN c = imag_i(gel(v,i));
     144       29789 :     if (gsigne(c) > 0) d += gtodouble(c);
     145             :   }
     146       17768 :   return d;
     147             : }
     148             : 
     149             : static long
     150       30047 : vgaell(GEN Vga)
     151             : {
     152       30047 :   if (lg(Vga) == 3)
     153       19071 :   { GEN c = gsub(gel(Vga,1), gel(Vga,2)); return gequal1(c) || gequalm1(c); }
     154       10976 :   return 0;
     155             : }
     156             : int
     157       71774 : Vgaeasytheta(GEN Vga) { return lg(Vga)-1 == 1 || vgaell(Vga); }
     158             : /* return b(n) := a(n) * n^c, when Vgaeasytheta(Vga) is set */
     159             : static GEN
     160       11004 : antwist(GEN an, GEN Vga, long prec)
     161             : {
     162             :   long l, i;
     163       11004 :   GEN b, c = vecmin(Vga);
     164       11004 :   if (gequal0(c)) return an;
     165        1764 :   l = lg(an); b = cgetg(l, t_VEC);
     166        1764 :   if (gequal1(c))
     167             :   {
     168        1036 :     if (typ(an) == t_VECSMALL)
     169        8631 :       for (i = 1; i < l; i++) gel(b,i) = mulss(an[i], i);
     170             :     else
     171       19551 :       for (i = 1; i < l; i++) gel(b,i) = gmulgu(gel(an,i), i);
     172             :   }
     173             :   else
     174             :   {
     175         728 :     GEN v = vecpowug(l-1, c, prec);
     176         728 :     if (typ(an) == t_VECSMALL)
     177           0 :       for (i = 1; i < l; i++) gel(b,i) = gmulsg(an[i], gel(v,i));
     178             :     else
     179       20965 :       for (i = 1; i < l; i++) gel(b,i) = gmul(gel(an,i), gel(v,i));
     180             :   }
     181        1764 :   return b;
     182             : }
     183             : 
     184             : static GEN
     185        6902 : theta_dual(GEN theta, GEN bn)
     186             : {
     187        6902 :   if (typ(bn)==t_INT) return NULL;
     188             :   else
     189             :   {
     190          77 :     GEN thetad = shallowcopy(theta), ldata = linit_get_ldata(theta);
     191          77 :     GEN Vga = ldata_get_gammavec(ldata);
     192          77 :     GEN tech = shallowcopy(linit_get_tech(theta));
     193          77 :     GEN an = theta_get_an(tech);
     194          77 :     long prec = nbits2prec(theta_get_bitprec(tech));
     195          77 :     GEN vb = ldata_vecan(bn, lg(an)-1, prec);
     196          77 :     if (!theta_get_m(tech) && Vgaeasytheta(Vga)) vb = antwist(vb, Vga, prec);
     197          77 :     gel(tech,1) = vb;
     198          77 :     gel(thetad,3) = tech; return thetad;
     199             :   }
     200             : }
     201             : 
     202             : static GEN
     203       37710 : domain_get_dom(GEN domain)  { return gel(domain,1); }
     204             : static long
     205       16560 : domain_get_der(GEN domain)  { return mael2(domain, 2, 1); }
     206             : static long
     207       22524 : domain_get_bitprec(GEN domain)  { return mael2(domain, 2, 2); }
     208             : GEN
     209       38235 : lfun_get_domain(GEN tech) { return gel(tech,1); }
     210             : long
     211          77 : lfun_get_bitprec(GEN tech){ return domain_get_bitprec(lfun_get_domain(tech)); }
     212             : GEN
     213           0 : lfun_get_dom(GEN tech) { return domain_get_dom(lfun_get_domain(tech)); }
     214             : 
     215             : GEN
     216        2035 : lfunprod_get_fact(GEN tech)  { return gel(tech, 2); }
     217             : 
     218             : GEN
     219       47239 : theta_get_an(GEN tdata)      { return gel(tdata, 1);}
     220             : GEN
     221        7133 : theta_get_K(GEN tdata)       { return gel(tdata, 2);}
     222             : GEN
     223        5935 : theta_get_R(GEN tdata)       { return gel(tdata, 3);}
     224             : long
     225       60608 : theta_get_bitprec(GEN tdata) { return itos(gel(tdata, 4));}
     226             : long
     227       91552 : theta_get_m(GEN tdata)       { return itos(gel(tdata, 5));}
     228             : GEN
     229       48611 : theta_get_tdom(GEN tdata)    { return gel(tdata, 6);}
     230             : GEN
     231       52601 : theta_get_isqrtN(GEN tdata)  { return gel(tdata, 7);}
     232             : 
     233             : /*******************************************************************/
     234             : /*  Helper functions related to Gamma products                     */
     235             : /*******************************************************************/
     236             : /* x != 0 */
     237             : static int
     238        5747 : serisscalar(GEN x)
     239             : {
     240             :   long i;
     241        5747 :   if (valser(x)) return 0;
     242        7448 :   for (i = lg(x)-1; i > 3; i--) if (!gequal0(gel(x,i))) return 0;
     243        5509 :   return 1;
     244             : }
     245             : 
     246             : /* return -itos(s) >= 0 if scalar s is (approximately) equal to a nonpositive
     247             :  * integer, and -1 otherwise */
     248             : static long
     249       14728 : isnegint(GEN s)
     250             : {
     251       14728 :   GEN r = ground(real_i(s));
     252       14728 :   if (signe(r) <= 0 && gequal(s, r)) return -itos(r);
     253       14602 :   return -1;
     254             : }
     255             : /* if s = a + O(x^n), a <= 0 integer, replace by a + b*x^n + O(x^(n+1)) */
     256             : static GEN
     257        5768 : serextendifnegint(GEN s, GEN b, long *ext)
     258             : {
     259        5768 :   if (!signe(s) || (serisscalar(s) && isnegint(gel(s,2)) >= 0))
     260             :   {
     261         112 :     long l = lg(s);
     262         112 :     GEN t = cgetg(l+1, t_SER);
     263         301 :     gel(t, l) = b; while (--l > 1) gel(t,l) = gel(s,l);
     264         112 :     if (gequal0(gel(t,2))) gel(t,2) = gen_0;
     265         112 :     t[1] = s[1]; s = normalizeser(t); *ext = 1;
     266             :   }
     267        5768 :   return s;
     268             : }
     269             : 
     270             : /* r/x + O(1), r != 0 */
     271             : static GEN
     272        3878 : serpole(GEN r)
     273             : {
     274        3878 :   GEN s = cgetg(3, t_SER);
     275        3878 :   s[1] = evalsigne(1)|evalvalser(-1)|evalvarn(0);
     276        3878 :   gel(s,2) = r; return s;
     277             : }
     278             : /* a0 +  a1 x + O(x^e), e >= 0 */
     279             : static GEN
     280        6706 : deg1ser_shallow(GEN a1, GEN a0, long v, long e)
     281        6706 : { return RgX_to_ser(deg1pol_shallow(a1, a0, v), e+2); }
     282             : 
     283             : /* pi^(-s/2) Gamma(s/2) */
     284             : static GEN
     285        8624 : gamma_R(GEN s, long *ext, long prec)
     286             : {
     287        8624 :   GEN s2 = gmul2n(s, -1);
     288             :   long ms;
     289             : 
     290        8624 :   if (typ(s) == t_SER)
     291        4018 :     s2 = serextendifnegint(s2, ghalf, ext);
     292        4606 :   else if ((ms = isnegint(s2)) >= 0)
     293             :   {
     294          35 :     GEN r = gmul(powPis(stoi(ms),prec), gdivsg(odd(ms)? -2: 2, mpfact(ms)));
     295          35 :     return serpole(r);
     296             :   }
     297        8589 :   return gdiv(ggamma(s2,prec), powPis(s2,prec));
     298             : }
     299             : /* gamma_R(s)gamma_R(s+1) = 2 (2pi)^(-s) Gamma(s) */
     300             : static GEN
     301        6363 : gamma_C(GEN s, long *ext, long prec)
     302             : {
     303             :   long ms;
     304        6363 :   if (typ(s) == t_SER)
     305        1750 :     s = serextendifnegint(s, gen_1, ext);
     306        4613 :   else if ((ms = isnegint(s)) >= 0)
     307             :   {
     308           0 :     GEN r = gmul(pow2Pis(stoi(ms),prec), gdivsg(odd(ms)? -2: 2, mpfact(ms)));
     309           0 :     return serpole(r);
     310             :   }
     311        6363 :   return gmul2n(gdiv(ggamma(s,prec), pow2Pis(s,prec)), 1);
     312             : }
     313             : 
     314             : static GEN
     315        1708 : gammafrac(GEN r, long d)
     316             : {
     317        1708 :   long i, l = labs(d) + 1, j = (d > 0)? 0: 2*d;
     318        1708 :   GEN T, v = cgetg(l, t_COL);
     319        4466 :   for (i = 1; i < l; i++, j += 2)
     320        2758 :     gel(v,i) = deg1pol_shallow(gen_1, gaddgs(r, j), 0);
     321        1708 :   T = RgV_prod(v); return d > 0? T: mkrfrac(gen_1, T);
     322             : }
     323             : 
     324             : /*
     325             : GR(s)=Pi^-(s/2)*gamma(s/2);
     326             : GC(s)=2*(2*Pi)^-s*gamma(s)
     327             : gdirect(F,s)=prod(i=1,#F,GR(s+F[i]))
     328             : gfact(F,s)=
     329             : { my([R,A,B]=gammafactor(F), [a,e]=A, [b,f]=B, p=poldegree(R));
     330             :   subst(R,x,s) * (2*Pi)^-p * prod(i=1,#a,GR(s+a[i])^e[i])
     331             :                            * prod(i=1,#b,GC(s+b[i])^f[i]); }
     332             : */
     333             : static GEN
     334       15225 : gammafactor(GEN Vga)
     335             : {
     336       15225 :   long i, r, c, l = lg(Vga);
     337       15225 :   GEN v, P, a, b, e, f, E, F = cgetg(l, t_VEC), R = gen_1;
     338       43001 :   for (i = 1; i < l; ++i)
     339             :   {
     340       27776 :     GEN a = gel(Vga,i), r = gmul2n(real_i(a), -1);
     341       27776 :     long q = itos(gfloor(r)); /* [Re a/2] */
     342       27776 :     r = gmul2n(gsubgs(r, q), 1);
     343       27776 :     gel(F,i) = gequal0(imag_i(a)) ? r : mkcomplex(r, imag_i(a)); /* 2{Re a/2} + I*(Im a) */
     344       27776 :     if (q) R = gmul(R, gammafrac(gel(F,i), q));
     345             :   }
     346       15225 :   F = vec_reduce(F, &E); l = lg(E);
     347       15225 :   v = cgetg(l, t_VEC);
     348       37772 :   for (i = 1; i < l; i++)
     349       22547 :       gel(v,i) = mkvec2(gsub(gel(F,i),gfloor(real_i(gel(F,i)))), stoi(E[i]));
     350       15225 :   gen_sort_inplace(v, (void*)cmp_universal, cmp_nodata, &P);
     351       15225 :   a = cgetg(l, t_VEC); e = cgetg(l, t_VECSMALL);
     352       15225 :   b = cgetg(l, t_VEC); f = cgetg(l, t_VECSMALL);
     353       31395 :   for (i = r = c = 1; i < l;)
     354       16170 :     if (i==l-1 || cmp_universal(gel(v,i), gel(v,i+1)))
     355        9793 :     { gel(a, r) = gel(F, P[i]); e[r++] = E[P[i]]; i++; }
     356             :     else
     357        6377 :     { gel(b, c) = gel(F, P[i]); f[c++] = E[P[i]]; i+=2; }
     358       15225 :   setlg(a, r); setlg(e, r);
     359       15225 :   setlg(b, c); setlg(f, c); return mkvec3(R, mkvec2(a,e), mkvec2(b,f));
     360             : }
     361             : 
     362             : static GEN
     363        3640 : polgammaeval(GEN F, GEN s)
     364             : {
     365        3640 :   GEN r = poleval(F, s);
     366        3640 :   if (typ(s) != t_SER && gequal0(r))
     367             :   { /* here typ(F) = t_POL */
     368             :     long e;
     369           7 :     for (e = 1;; e++)
     370             :     {
     371           7 :       F = RgX_deriv(F); r = poleval(F,s);
     372           7 :       if (!gequal0(r)) break;
     373             :     }
     374           7 :     if (e > 1) r = gdiv(r, mpfact(e));
     375           7 :     r = serpole(r); setvalser(r, e);
     376             :   }
     377        3640 :   return r;
     378             : }
     379             : static long
     380        1799 : rfrac_degree(GEN R)
     381             : {
     382        1799 :   GEN a = gel(R,1), b = gel(R,2);
     383        1799 :   return ((typ(a) == t_POL)? degpol(a): 0) - degpol(b);
     384             : }
     385             : static GEN
     386       14000 : fracgammaeval(GEN F, GEN s, long prec)
     387             : {
     388       14000 :   GEN R = gel(F,1);
     389             :   long d;
     390       14000 :   switch(typ(R))
     391             :   {
     392          42 :     case t_POL:
     393          42 :       d = degpol(R);
     394          42 :       R = polgammaeval(R, s); break;
     395        1799 :     case t_RFRAC:
     396        1799 :       d = rfrac_degree(R);
     397        1799 :       R = gdiv(polgammaeval(gel(R,1), s), polgammaeval(gel(R,2), s)); break;
     398       12159 :     default: return R;
     399             :   }
     400        1841 :   return gmul(R, powrs(Pi2n(1,prec), -d));
     401             : }
     402             : 
     403             : static GEN
     404       14000 : gammafactproduct(GEN F, GEN s, long *ext, long prec)
     405             : {
     406       14000 :   pari_sp av = avma;
     407       14000 :   GEN R = gel(F,2), Rw = gel(R,1), Re = gel(R,2);
     408       14000 :   GEN C = gel(F,3), Cw = gel(C,1), Ce = gel(C,2), z = fracgammaeval(F,s,prec);
     409       14000 :   long i, lR = lg(Rw), lC = lg(Cw);
     410       14000 :   *ext = 0;
     411       22624 :   for (i = 1; i < lR; i++)
     412        8624 :     z = gmul(z, gpowgs(gamma_R(gadd(s,gel(Rw, i)), ext, prec), Re[i]));
     413       20363 :   for (i = 1; i < lC; i++)
     414        6363 :     z = gmul(z, gpowgs(gamma_C(gadd(s,gel(Cw, i)), ext, prec), Ce[i]));
     415       14000 :   return gerepileupto(av, z);
     416             : }
     417             : 
     418             : static int
     419        4739 : gammaordinary(GEN Vga, GEN s)
     420             : {
     421        4739 :   long i, d = lg(Vga)-1;
     422       12985 :   for (i = 1; i <= d; i++)
     423             :   {
     424        8337 :     GEN z = gadd(s, gel(Vga,i));
     425             :     long e;
     426        8337 :     if (gexpo(imag_i(z)) < -10)
     427             :     {
     428        8337 :       z = real_i(z);
     429        8337 :       if (gsigne(z) <= 0) { (void)grndtoi(z, &e); if (e < -10) return 0; }
     430             :     }
     431             :   }
     432        4648 :   return 1;
     433             : }
     434             : 
     435             : /* Exponent A of t in asymptotic expansion; K(t) ~ C t^A exp(-pi d t^(2/d)).
     436             :  * suma = vecsum(Vga)*/
     437             : static double
     438       74457 : gammavec_expo(long d, double suma) { return (1 - d + suma) / d; }
     439             : 
     440             : /*******************************************************************/
     441             : /*       First part: computations only involving Theta(t)          */
     442             : /*******************************************************************/
     443             : 
     444             : static void
     445      118747 : get_cone(GEN t, double *r, double *a)
     446             : {
     447      118747 :   const long prec = LOWDEFAULTPREC;
     448      118747 :   if (typ(t) == t_COMPLEX)
     449             :   {
     450       21098 :     t  = gprec_w(t, prec);
     451       21098 :     *r = gtodouble(gabs(t, prec));
     452       21098 :     *a = fabs(gtodouble(garg(t, prec)));
     453             :   }
     454             :   else
     455             :   {
     456       97649 :     *r = fabs(gtodouble(t));
     457       97649 :     *a = 0.;
     458             :   }
     459      118747 :   if (!*r && !*a) pari_err_DOMAIN("lfunthetainit","t","=",gen_0,t);
     460      118740 : }
     461             : /* slightly larger cone than necessary, to avoid round-off problems */
     462             : static void
     463       70136 : get_cone_fuzz(GEN t, double *r, double *a)
     464       70136 : { get_cone(t, r, a); *r -= 1e-10; if (*a) *a += 1e-10; }
     465             : 
     466             : /* Initialization m-th Theta derivative. tdom is either
     467             :  * - [rho,alpha]: assume |t| >= rho and |arg(t)| <= alpha
     468             :  * - a positive real scalar: assume t real, t >= tdom;
     469             :  * - a complex number t: compute at t;
     470             :  * N is the conductor (either the true one from ldata or a guess from
     471             :  * lfunconductor) */
     472             : long
     473       56696 : lfunthetacost(GEN ldata, GEN tdom, long m, long bitprec)
     474             : {
     475       56696 :   pari_sp av = avma;
     476       56696 :   GEN Vga = ldata_get_gammavec(ldata);
     477       56696 :   long d = lg(Vga)-1;
     478       56696 :   double k1 = maxdd(ldata_get_k1_dbl(ldata), 0.);
     479       56696 :   double c = d/2., a, A, B, logC, al, rho, T;
     480       56696 :   double N = gtodouble(ldata_get_conductor(ldata));
     481             : 
     482       56696 :   if (!N) pari_err_TYPE("lfunthetaneed [missing conductor]", ldata);
     483       56696 :   if (typ(tdom) == t_VEC && lg(tdom) == 3)
     484             :   {
     485           7 :     rho= gtodouble(gel(tdom,1));
     486           7 :     al = gtodouble(gel(tdom,2));
     487             :   }
     488             :   else
     489       56689 :     get_cone_fuzz(tdom, &rho, &al);
     490       56689 :   A = gammavec_expo(d, gtodouble(sumVga(Vga))); set_avma(av);
     491       56689 :   a = (A+k1+1) + (m-1)/c;
     492       56689 :   if (fabs(a) < 1e-10) a = 0.;
     493       56689 :   logC = c*M_LN2 - log(c)/2;
     494             :   /* +1: fudge factor */
     495       56689 :   B = M_LN2*bitprec+logC+m*log(2*M_PI) + 1 + (k1+1)*log(N)/2 - (k1+m+1)*log(rho);
     496       56689 :   if (al)
     497             :   { /* t = rho e^(i*al), T^(1/c) = Re(t^(1/c)) > 0, T = rho cos^c(al/c) */
     498       10556 :     double z = cos(al/c);
     499       10556 :     if (z <= 0)
     500           7 :       pari_err_DOMAIN("lfunthetaneed", "arg t", ">", dbltor(c*M_PI/2), tdom);
     501       10549 :     T = (d == 2 && typ(tdom) != t_VEC)? gtodouble(real_i(tdom)): rho*pow(z,c);
     502       10549 :     B -= log(z) * (c * (k1+A+1) + m);
     503             :   }
     504             :   else
     505       46133 :     T = rho;
     506       56682 :   if (B <= 0) return 0;
     507       56682 :   A = floor(0.9 + dblcoro526(a,c,B) / T * sqrt(N));
     508       56682 :   if (dblexpo(A) >= BITS_IN_LONG-1) pari_err_OVERFLOW("lfunthetacost");
     509       56675 :   return (long)A;
     510             : }
     511             : long
     512          21 : lfunthetacost0(GEN L, GEN tdom, long m, long bitprec)
     513             : {
     514             :   long n;
     515          21 :   if (is_linit(L) && linit_get_type(L)==t_LDESC_THETA)
     516           7 :   {
     517           7 :     GEN tech = linit_get_tech(L);
     518           7 :     n = lg(theta_get_an(tech))-1;
     519             :   }
     520             :   else
     521             :   {
     522          14 :     pari_sp av = avma;
     523          14 :     GEN ldata = lfunmisc_to_ldata_shallow(L);
     524          14 :     n = lfunthetacost(ldata, tdom? tdom: gen_1, m, bitprec);
     525           7 :     set_avma(av);
     526             :   }
     527          14 :   return n;
     528             : }
     529             : 
     530             : static long
     531        5544 : fracgammadegree(GEN FVga)
     532        5544 : { GEN F = gel(FVga,1); return (typ(F)==t_RFRAC)? degpol(gel(F,2)): 0; }
     533             : 
     534             : /* Poles of a L-function can be represented in the following ways:
     535             :  * 1) Nothing (ldata has only 6 components, ldata_get_residue = NULL).
     536             :  * 2) a complex number (single pole at s = k with given residue, unknown if 0).
     537             :  * 3) A vector (possibly empty) of 2-component vectors [a, ra], where a is the
     538             :  * pole, ra a t_SER: its Taylor expansion at a. A t_VEC encodes the polar
     539             :  * part of L, a t_COL, the polar part of Lambda */
     540             : 
     541             : /* 'a' a complex number (pole), 'r' the polar part of L at 'a';
     542             :  * return 'R' the polar part of Lambda at 'a' */
     543             : static GEN
     544        3920 : rtoR(GEN a, GEN r, GEN FVga, GEN N, long prec)
     545             : {
     546        3920 :   long v = lg(r)-2, d = fracgammadegree(FVga), ext;
     547        3920 :   GEN Na, as = deg1ser_shallow(gen_1, a, varn(r), v);
     548        3920 :   Na = gpow(N, gdivgu(as, 2), prec);
     549             :   /* make up for a possible loss of accuracy */
     550        3920 :   if (d) as = deg1ser_shallow(gen_1, a, varn(r), v + d);
     551        3920 :   return gmul(gmul(r, Na), gammafactproduct(FVga, as, &ext, prec));
     552             : }
     553             : 
     554             : /* assume r in normalized form: t_VEC of pairs [be,re] */
     555             : GEN
     556        3689 : lfunrtopoles(GEN r)
     557             : {
     558        3689 :   long j, l = lg(r);
     559        3689 :   GEN v = cgetg(l, t_VEC);
     560        7609 :   for (j = 1; j < l; j++)
     561             :   {
     562        3920 :     GEN rj = gel(r,j), a = gel(rj,1);
     563        3920 :     gel(v,j) = a;
     564             :   }
     565        3689 :   gen_sort_inplace(v, (void*)&cmp_universal, cmp_nodata, NULL);
     566        3689 :   return v;
     567             : }
     568             : 
     569             : /* r / x + O(1) */
     570             : static GEN
     571        3969 : simple_pole(GEN r)
     572        3969 : { return isintzero(r)? gen_0: serpole(r); }
     573             : static GEN
     574        4809 : normalize_simple_pole(GEN r, GEN k)
     575             : {
     576        4809 :   long tx = typ(r);
     577        4809 :   if (is_vec_t(tx)) return r;
     578        3969 :   if (!is_scalar_t(tx)) pari_err_TYPE("lfunrootres [poles]", r);
     579        3969 :   return mkvec(mkvec2(k, simple_pole(r)));
     580             : }
     581             : /* normalize the description of a polar part */
     582             : static GEN
     583        4536 : normalizepoles(GEN r, GEN k)
     584             : {
     585             :   long iv, j, l;
     586             :   GEN v;
     587        4536 :   if (!is_vec_t(typ(r))) return normalize_simple_pole(r, k);
     588        2163 :   v = cgetg_copy(r, &l);
     589        5425 :   for (j = iv = 1; j < l; j++)
     590             :   {
     591        3262 :     GEN rj = gel(r,j), a = gel(rj,1), ra = gel(rj,2);
     592        3262 :     if (!is_scalar_t(typ(a)) || typ(ra) != t_SER)
     593           0 :       pari_err_TYPE("lfunrootres [poles]",r);
     594        3262 :     if (valser(ra) >= 0) continue;
     595        3248 :     gel(v,iv++) = rj;
     596             :   }
     597        2163 :   setlg(v, iv); return v;
     598             : }
     599             : static int
     600        7371 : residues_known(GEN r)
     601             : {
     602        7371 :   long i, l = lg(r);
     603        7371 :   if (isintzero(r)) return 0;
     604        7070 :   if (!is_vec_t(typ(r))) return 1;
     605        9968 :   for (i = 1; i < l; i++)
     606             :   {
     607        6167 :     GEN ri = gel(r,i);
     608        6167 :     if (!is_vec_t(typ(ri)) || lg(ri)!=3)
     609           0 :       pari_err_TYPE("lfunrootres [poles]",r);
     610        6167 :     if (isintzero(gel(ri, 2))) return 0;
     611             :   }
     612        3801 :   return 1;
     613             : }
     614             : 
     615             : /* Compute R's from r's (r = Taylor devts of L(s), R of Lambda(s)).
     616             :  * 'r/eno' passed to override the one from ldata  */
     617             : static GEN
     618       15568 : lfunrtoR_i(GEN ldata, GEN r, GEN eno, long prec)
     619             : {
     620       15568 :   GEN Vga = ldata_get_gammavec(ldata), N = ldata_get_conductor(ldata);
     621             :   GEN R, vr, FVga;
     622       15568 :   pari_sp av = avma;
     623             :   long lr, j, jR;
     624       15568 :   GEN k = ldata_get_k(ldata);
     625             : 
     626       15568 :   if (!r || isintzero(eno) || !residues_known(r))
     627       11032 :     return gen_0;
     628        4536 :   r = normalizepoles(r, k);
     629        4536 :   if (typ(r) == t_COL) return gerepilecopy(av, r);
     630        3689 :   if (typ(ldata_get_dual(ldata)) != t_INT)
     631           0 :     pari_err(e_MISC,"please give the Taylor development of Lambda");
     632        3689 :   vr = lfunrtopoles(r); lr = lg(vr);
     633        3689 :   FVga = gammafactor(Vga);
     634        3689 :   R = cgetg(2*lr, t_COL);
     635        7609 :   for (j = jR = 1; j < lr; j++)
     636             :   {
     637        3920 :     GEN rj = gel(r,j), a = gel(rj,1), ra = gel(rj,2);
     638        3920 :     GEN Ra = rtoR(a, ra, FVga, N, prec);
     639        3920 :     GEN b = gsub(k, conj_i(a));
     640        3920 :     if (lg(Ra)-2 < -valser(Ra))
     641           0 :       pari_err(e_MISC,
     642             :         "please give more terms in L function's Taylor development at %Ps", a);
     643        3920 :     gel(R,jR++) = mkvec2(a, Ra);
     644        3920 :     if (!tablesearch(vr, b, (int (*)(GEN,GEN))&cmp_universal))
     645             :     {
     646        3731 :       GEN mX = gneg(pol_x(varn(Ra)));
     647        3731 :       GEN Rb = gmul(eno, gsubst(conj_i(Ra), varn(Ra), mX));
     648        3731 :       gel(R,jR++) = mkvec2(b, Rb);
     649             :     }
     650             :   }
     651        3689 :   setlg(R, jR); return gerepilecopy(av, R);
     652             : }
     653             : static GEN
     654       15106 : lfunrtoR_eno(GEN ldata, GEN eno, long prec)
     655       15106 : { return lfunrtoR_i(ldata, ldata_get_residue(ldata), eno, prec); }
     656             : static GEN
     657       13454 : lfunrtoR(GEN ldata, long prec)
     658       13454 : { return lfunrtoR_eno(ldata, ldata_get_rootno(ldata), prec); }
     659             : 
     660             : static long
     661       13454 : prec_fix(long prec)
     662             : {
     663             : #ifndef LONG_IS_64BIT
     664             :   /* make sure that default accuracy is the same on 32/64bit */
     665        1922 :   if (odd(prec)) prec += EXTRAPREC64;
     666             : #endif
     667       13454 :   return prec;
     668             : }
     669             : 
     670             : /* thetainit using {an: n <= L}; if (m = 0 && easytheta), an2 is an * n^al */
     671             : static GEN
     672       13454 : lfunthetainit0(GEN ldata, GEN tdom, GEN an2, long m,
     673             :     long bitprec, long extrabit)
     674             : {
     675       13454 :   long prec = nbits2prec(bitprec);
     676       13454 :   GEN tech, N = ldata_get_conductor(ldata);
     677       13454 :   GEN K = gammamellininvinit(ldata, m, bitprec + extrabit);
     678       13454 :   GEN R = lfunrtoR(ldata, prec);
     679       13454 :   if (!tdom) tdom = gen_1;
     680       13454 :   if (typ(tdom) != t_VEC)
     681             :   {
     682             :     double r, a;
     683       13447 :     get_cone_fuzz(tdom, &r, &a);
     684       13447 :     tdom = mkvec2(dbltor(r), a? dbltor(a): gen_0);
     685             :   }
     686       13454 :   prec += maxss(EXTRAPREC64, nbits2extraprec(extrabit));
     687       13454 :   tech = mkvecn(7, an2,K,R, stoi(bitprec), stoi(m), tdom,
     688             :                    gsqrt(ginv(N), prec_fix(prec)));
     689       13454 :   return mkvec3(mkvecsmall(t_LDESC_THETA), ldata, tech);
     690             : }
     691             : 
     692             : /* tdom: 1) positive real number r, t real, t >= r; or
     693             :  *       2) [r,a], describing the cone |t| >= r, |arg(t)| <= a */
     694             : static GEN
     695        7812 : lfunthetainit_i(GEN data, GEN tdom, long m, long bit)
     696             : {
     697        7812 :   GEN ldata = lfunmisc_to_ldata_shallow(data);
     698        7812 :   long b = 32, L = lfunthetacost(ldata, tdom, m, bit), prec = nbits2prec(bit);
     699        7798 :   GEN ldatan = ldata_newprec(ldata, prec);
     700        7798 :   GEN an = ldata_vecan(ldata_get_an(ldatan), L, prec);
     701        7798 :   GEN Vga = ldata_get_gammavec(ldatan);
     702        7798 :   if (m == 0 && Vgaeasytheta(Vga)) an = antwist(an, Vga, prec);
     703        7798 :   if (typ(an) != t_VECSMALL) b = maxss(b, gexpo(an));
     704        7798 :   return lfunthetainit0(ldatan, tdom, an, m, bit, b);
     705             : }
     706             : 
     707             : GEN
     708         322 : lfunthetainit(GEN ldata, GEN tdom, long m, long bitprec)
     709             : {
     710         322 :   pari_sp av = avma;
     711         322 :   GEN S = lfunthetainit_i(ldata, tdom? tdom: gen_1, m, bitprec);
     712         322 :   return gerepilecopy(av, S);
     713             : }
     714             : 
     715             : GEN
     716        1274 : lfunan(GEN ldata, long L, long prec)
     717             : {
     718        1274 :   pari_sp av = avma;
     719             :   GEN an ;
     720        1274 :   ldata = ldata_newprec(lfunmisc_to_ldata_shallow(ldata), prec);
     721        1274 :   an = gerepilecopy(av, ldata_vecan(ldata_get_an(ldata), L, prec));
     722        1218 :   if (typ(an) != t_VEC) an = vecsmall_to_vec_inplace(an);
     723        1218 :   return an;
     724             : }
     725             : 
     726             : static GEN
     727       10773 : mulrealvec(GEN x, GEN y)
     728             : {
     729       10773 :   if (is_vec_t(typ(x)) && is_vec_t(typ(y)))
     730          84 :     pari_APPLY_same(mulreal(gel(x,i),gel(y,i)))
     731             :   else
     732       10745 :     return mulreal(x,y);
     733             : }
     734             : static GEN
     735       21199 : gmulvec(GEN x, GEN y)
     736             : {
     737       21199 :   if (is_vec_t(typ(x)) && is_vec_t(typ(y)))
     738        1974 :     pari_APPLY_same(gmul(gel(x,i),gel(y,i)))
     739             :   else
     740       20674 :     return gmul(x,y);
     741             : }
     742             : static GEN
     743        6881 : gdivvec(GEN x, GEN y)
     744             : {
     745        6881 :   if (is_vec_t(typ(x)) && is_vec_t(typ(y)))
     746        1498 :     pari_APPLY_same(gdiv(gel(x,i),gel(y,i)))
     747             :   else
     748        6454 :     return gdiv(x,y);
     749             : }
     750             : 
     751             : static GEN
     752        3486 : gsubvec(GEN x, GEN y)
     753             : {
     754        3486 :   if (is_vec_t(typ(x)) && !is_vec_t(typ(y)))
     755           0 :     pari_APPLY_same(gsub(gel(x,i),y))
     756             :   else
     757        3486 :     return gsub(x,y);
     758             : }
     759             : 
     760             : /* return [1^(2/d), 2^(2/d),...,lim^(2/d)] */
     761             : static GEN
     762        7133 : mkvroots(long d, long lim, long prec)
     763             : {
     764        7133 :   if (d <= 4)
     765             :   {
     766        6783 :     GEN v = cgetg(lim+1,t_VEC);
     767             :     long n;
     768        6783 :     switch(d)
     769             :     {
     770        2072 :       case 1:
     771       38204 :         for (n=1; n <= lim; n++) gel(v,n) = sqru(n);
     772        2072 :         return v;
     773        1267 :       case 2:
     774      227213 :         for (n=1; n <= lim; n++) gel(v,n) = utoipos(n);
     775        1267 :         return v;
     776        1983 :       case 4:
     777     6079017 :         for (n=1; n <= lim; n++) gel(v,n) = sqrtr(utor(n, prec));
     778        1983 :         return v;
     779             :     }
     780             :   }
     781        1811 :   return vecpowug(lim, gdivgu(gen_2,d), prec);
     782             : }
     783             : 
     784             : GEN
     785       54169 : lfunthetacheckinit(GEN data, GEN t, long m, long bitprec)
     786             : {
     787       54169 :   if (is_linit(data) && linit_get_type(data)==t_LDESC_THETA)
     788             :   {
     789       48611 :     GEN tdom, thetainit = linit_get_tech(data);
     790       48611 :     long bitprecnew = theta_get_bitprec(thetainit);
     791       48611 :     long m0 = theta_get_m(thetainit);
     792             :     double r, al, rt, alt;
     793       48611 :     if (m0 != m)
     794           0 :       pari_err_DOMAIN("lfuntheta","derivative order","!=", stoi(m),stoi(m0));
     795       48611 :     if (bitprec > bitprecnew) goto INIT;
     796       48611 :     get_cone(t, &rt, &alt);
     797       48611 :     tdom = theta_get_tdom(thetainit);
     798       48611 :     r = gtodouble(gel(tdom,1));
     799       48611 :     al= gtodouble(gel(tdom,2)); if (rt >= r && alt <= al) return data;
     800             :   }
     801        5558 : INIT:
     802        7399 :   return lfunthetainit_i(data, t, m, bitprec);
     803             : }
     804             : 
     805             : static GEN
     806    14711879 : get_an(GEN an, long n)
     807             : {
     808    14711879 :   if (typ(an) == t_VECSMALL) { long a = an[n]; if (a) return stoi(a); }
     809    14711879 :   else { GEN a = gel(an,n); if (a && !gequal0(a)) return a; }
     810    12686150 :   return NULL;
     811             : }
     812             : /* x * an[n] */
     813             : static GEN
     814    11137294 : mul_an(GEN an, long n, GEN x)
     815             : {
     816    11137294 :   if (typ(an) == t_VECSMALL) { long a = an[n]; if (a) return gmulsg(a,x); }
     817     5750325 :   else { GEN a = gel(an,n); if (a && !gequal0(a)) return gmul(a,x); }
     818     2293863 :   return NULL;
     819             : }
     820             : /* 2*t^a * x **/
     821             : static GEN
     822      191210 : mulT(GEN t, GEN a, GEN x, long prec)
     823             : {
     824      191210 :   if (gequal0(a)) return gmul2n(x,1);
     825       17899 :   return gmul(x, gmul2n(gequal1(a)? t: gpow(t,a,prec), 1));
     826             : }
     827             : 
     828             : static GEN
     829    28300971 : vecan_cmul(void *E, GEN P, long a, GEN x)
     830             : {
     831             :   (void)E;
     832    28300971 :   if (typ(P) == t_VECSMALL)
     833    23772460 :     return (a==0 || !P[a])? NULL: gmulsg(P[a], x);
     834             :   else
     835     4528511 :     return (a==0 || !gel(P,a))? NULL: gmul(gel(P,a), x);
     836             : }
     837             : /* d=2, 2 sum_{n <= N} a(n) (n t)^al q^n, q = exp(-2pi t),
     838             :  * an2[n] = a(n) * n^al */
     839             : static GEN
     840      154038 : theta2_i(GEN an2, long N, GEN t, GEN al, long prec)
     841             : {
     842      154038 :   GEN S, q, pi2 = Pi2n(1,prec);
     843      154036 :   const struct bb_algebra *alg = get_Rg_algebra();
     844      154034 :   setsigne(pi2,-1); q = gexp(gmul(pi2, t), prec);
     845             :   /* Brent-Kung in case the a_n are small integers */
     846      154042 :   S = gen_bkeval(an2, N, q, 1, NULL, alg, vecan_cmul);
     847      154040 :   return mulT(t, al, S, prec);
     848             : }
     849             : static GEN
     850      148345 : theta2(GEN an2, long N, GEN t, GEN al, long prec)
     851             : {
     852      148345 :   pari_sp av = avma;
     853      148345 :   return gerepileupto(av, theta2_i(an2, N, t, al, prec));
     854             : }
     855             : 
     856             : /* d=1, 2 sum_{n <= N} a_n (n t)^al q^(n^2), q = exp(-pi t^2),
     857             :  * an2[n] is a_n n^al */
     858             : static GEN
     859       37170 : theta1(GEN an2, long N, GEN t, GEN al, long prec)
     860             : {
     861       37170 :   GEN q = gexp(gmul(negr(mppi(prec)), gsqr(t)), prec);
     862       37170 :   GEN vexp = gsqrpowers(q, N), S = gen_0;
     863       37170 :   pari_sp av = avma;
     864             :   long n;
     865     6748474 :   for (n = 1; n <= N; n++)
     866             :   {
     867     6711304 :     GEN c = mul_an(an2, n, gel(vexp,n));
     868     6711304 :     if (c)
     869             :     {
     870     5644015 :       S = gadd(S, c);
     871     5644015 :       if (gc_needed(av, 3)) S = gerepileupto(av, S);
     872             :     }
     873             :   }
     874       37170 :   return mulT(t, al, S, prec);
     875             : }
     876             : 
     877             : /* If m > 0, compute m-th derivative of theta(t) = theta0(t/sqrt(N))
     878             :  * with absolute error 2^-bitprec; theta(t)=\sum_{n\ge1}a(n)K(nt/N^(1/2)) */
     879             : GEN
     880       46952 : lfuntheta(GEN data, GEN t, long m, long bitprec)
     881             : {
     882       46952 :   pari_sp ltop = avma;
     883             :   long limt, d;
     884             :   GEN isqN, vecan, Vga, ldata, theta, thetainit, S;
     885             :   long n, prec;
     886             : 
     887       46952 :   theta = lfunthetacheckinit(data, t, m, bitprec);
     888       46945 :   ldata = linit_get_ldata(theta);
     889       46945 :   thetainit = linit_get_tech(theta);
     890       46945 :   vecan = theta_get_an(thetainit);
     891       46945 :   isqN = theta_get_isqrtN(thetainit);
     892       46945 :   prec = maxss(realprec(isqN), nbits2prec(bitprec));
     893       46945 :   t = gprec_w(t, prec);
     894       46945 :   limt = lg(vecan)-1;
     895       46945 :   if (theta == data)
     896       45111 :     limt = minss(limt, lfunthetacost(ldata, t, m, bitprec));
     897       46945 :   if (!limt)
     898             :   {
     899          14 :     set_avma(ltop); S = real_0_bit(-bitprec);
     900          14 :     if (!is_real_t(typ(t)) || !ldata_isreal(ldata))
     901           7 :       S = gerepilecopy(ltop, mkcomplex(S,S));
     902          14 :     return S;
     903             :   }
     904       46931 :   t = gmul(t, isqN);
     905       46931 :   Vga = ldata_get_gammavec(ldata);
     906       46931 :   d = lg(Vga)-1;
     907       46931 :   if (m == 0 && Vgaeasytheta(Vga))
     908             :   {
     909       42864 :     if (theta_get_m(thetainit) > 0) vecan = antwist(vecan, Vga, prec);
     910       42864 :     if (d == 1) S = theta1(vecan, limt, t, gel(Vga,1), prec);
     911        5694 :     else        S = theta2_i(vecan, limt, t, vecmin(Vga), prec);
     912             :   }
     913             :   else
     914             :   {
     915        4067 :     GEN K = theta_get_K(thetainit);
     916        4067 :     GEN vroots = mkvroots(d, limt, prec);
     917             :     pari_sp av;
     918        4067 :     t = gpow(t, gdivgu(gen_2,d), prec);
     919        4067 :     S = gen_0; av = avma;
     920    14715946 :     for (n = 1; n <= limt; ++n)
     921             :     {
     922    14711879 :       GEN nt, an = get_an(vecan, n);
     923    14711879 :       if (!an) continue;
     924     2025729 :       nt = gmul(gel(vroots,n), t);
     925     2025729 :       if (m) an = gmul(an, powuu(n, m));
     926     2025729 :       S = gadd(S, gmul(an, gammamellininvrt(K, nt, bitprec)));
     927     2025729 :       if ((n & 0x1ff) == 0) S = gerepileupto(av, S);
     928             :     }
     929        4067 :     if (m) S = gmul(S, gpowgs(isqN, m));
     930             :   }
     931       46931 :   return gerepileupto(ltop, S);
     932             : }
     933             : 
     934             : /*******************************************************************/
     935             : /* Second part: Computation of L-Functions.                        */
     936             : /*******************************************************************/
     937             : 
     938             : struct lfunp {
     939             :   long precmax, Dmax, D, M, m0, nmax, d, vgaell;
     940             :   double k1, dc, dw, dh, MAXs, sub;
     941             :   GEN L, an, bn;
     942             : };
     943             : 
     944             : static void
     945       17768 : lfunp_set(GEN ldata, long der, long bitprec, struct lfunp *S)
     946             : {
     947       17768 :   const long derprec = (der > 1)? dbllog2(mpfact(der)): 0; /* log2(der!) */
     948             :   GEN Vga, N, L, k;
     949             :   long k1, d, m, M, flag, nmax;
     950             :   double a, A, E, hd, Ep, d2, suma, maxs, mins, sub, B0,B1;
     951             :   double logN2, logC, Lestimate, Mestimate;
     952             : 
     953       17768 :   Vga = ldata_get_gammavec(ldata);
     954       17768 :   S->d = d = lg(Vga)-1; d2 = d/2.;
     955             : 
     956       17768 :   suma = gtodouble(sumVga(Vga));
     957       17768 :   k = ldata_get_k(ldata);
     958       17768 :   N = ldata_get_conductor(ldata);
     959       17768 :   logN2 = log(gtodouble(N)) / 2;
     960       17768 :   maxs = S->dc + S->dw;
     961       17768 :   mins = S->dc - S->dw;
     962       17768 :   S->MAXs = maxdd(maxs, gtodouble(k)-mins);
     963             : 
     964             :   /* we compute Lambda^(der)(s) / der!; need to compensate for L^(der)(s)
     965             :    * ln |gamma(s)| ~ -(pi/4) \sum_i |Im(s + a_i)|; max with 1: fudge factor */
     966       17768 :   a = (M_PI/(4*M_LN2))*(d*S->dh + sumVgaimpos(Vga));
     967       17768 :   S->D = (long)ceil(bitprec + derprec + maxdd(a, 1));
     968       17768 :   E = M_LN2*S->D; /* D:= required absolute bitprec */
     969             : 
     970       17768 :   Ep = E + maxdd(M_PI * S->dh * d2, (d*S->MAXs + suma - 1) * log(E));
     971       17768 :   hd = d2*M_PI*M_PI / Ep;
     972       17768 :   S->m0 = (long)ceil(M_LN2/hd);
     973       17768 :   hd = M_LN2/S->m0;
     974             : 
     975       17768 :   logC = d2*M_LN2 - log(d2)/2;
     976       17768 :   k1 = maxdd(ldata_get_k1_dbl(ldata), 0.);
     977       17768 :   S->k1 = k1; /* assume |a_n| << n^k1 with small implied constant */
     978       17768 :   A = gammavec_expo(d, suma);
     979             : 
     980       17768 :   sub = 0.;
     981       17768 :   if (mins > 1)
     982             :   {
     983        4739 :     GEN sig = dbltor(mins);
     984        4739 :     sub += logN2*mins;
     985        4739 :     if (gammaordinary(Vga, sig))
     986             :     {
     987             :       long ext;
     988        4648 :       GEN gas = gammafactproduct(gammafactor(Vga), sig, &ext, LOWDEFAULTPREC);
     989        4648 :       if (typ(gas) != t_SER)
     990             :       {
     991        4648 :         double dg = dbllog2(gas);
     992        4648 :         if (dg > 0) sub += dg * M_LN2;
     993             :       }
     994             :     }
     995             :   }
     996       17768 :   S->sub = sub;
     997       17768 :   M = 1000;
     998       17768 :   L = cgetg(M+2, t_VECSMALL);
     999       17768 :   a = S->k1 + A;
    1000             : 
    1001       17768 :   B0 = 5 + E - S->sub + logC + S->k1*logN2; /* 5 extra bits */
    1002       17768 :   B1 = hd * (S->MAXs - S->k1);
    1003       17768 :   Lestimate = dblcoro526(a + S->MAXs - 2./d, d/2.,
    1004       17768 :     E - S->sub + logC - log(2*M_PI*hd) + S->MAXs*logN2);
    1005       17768 :   Mestimate = ((Lestimate > 0? log(Lestimate): 0) + logN2) / hd;
    1006       17768 :   nmax = 0;
    1007       17768 :   flag = 0;
    1008       17768 :   for (m = 0;; m++)
    1009     1657510 :   {
    1010     1675278 :     double x, H = logN2 - m*hd, B = B0 + m*B1;
    1011             :     long n;
    1012     1675278 :     x = dblcoro526(a, d/2., B);
    1013     1675278 :     n = floor(x*exp(H));
    1014     1675278 :     if (n > nmax) nmax = n;
    1015     1675278 :     if (m > M) { M *= 2; L = vecsmall_lengthen(L,M+2); }
    1016     1675278 :     L[m+1] = n;
    1017     1675278 :     if (n == 0) { if (++flag > 2 && m > Mestimate) break; } else flag = 0;
    1018             :   }
    1019       18608 :   m -= 2; while (m > 0 && !L[m]) m--;
    1020       17768 :   if (m == 0) { nmax = 1; L[1] = 1; m = 1; } /* can happen for tiny bitprec */
    1021       17768 :   setlg(L, m+1); S->M = m-1;
    1022       17768 :   S->L = L;
    1023       17768 :   S->nmax = nmax;
    1024             : 
    1025       17768 :   S->Dmax = S->D + (long)ceil((S->M * hd * S->MAXs - S->sub) / M_LN2);
    1026       17768 :   if (S->Dmax < S->D) S->Dmax = S->D;
    1027       17768 :   S->precmax = nbits2prec(S->Dmax);
    1028       17768 :   if (DEBUGLEVEL > 1)
    1029           0 :     err_printf("Dmax=%ld, D=%ld, M = %ld, nmax = %ld, m0 = %ld\n",
    1030             :                S->Dmax,S->D,S->M,S->nmax, S->m0);
    1031       17768 : }
    1032             : 
    1033             : static GEN
    1034        5810 : lfuninit_pol(GEN v, GEN poqk, long prec)
    1035             : {
    1036        5810 :   long m, M = lg(v) - 2;
    1037        5810 :   GEN pol = cgetg(M+3, t_POL);
    1038        5810 :   pol[1] = evalsigne(1) | evalvarn(0);
    1039        5810 :   gel(pol, 2) = gprec_w(gmul2n(gel(v,1), -1), prec);
    1040        5810 :   if (poqk)
    1041      301158 :     for (m = 2; m <= M+1; m++)
    1042      295404 :       gel(pol, m+1) = gprec_w(gmul(gel(poqk,m), gel(v,m)), prec);
    1043             :   else
    1044        2324 :     for (m = 2; m <= M+1; m++)
    1045        2268 :       gel(pol, m+1) = gprec_w(gel(v,m), prec);
    1046        5810 :   return RgX_renormalize_lg(pol, M+3);
    1047             : }
    1048             : 
    1049             : static void
    1050       51803 : worker_init(long q, GEN *an, GEN *bn, GEN *AB, GEN *A, GEN *B)
    1051             : {
    1052       51803 :   if (typ(*bn) == t_INT) *bn = NULL;
    1053       51803 :   if (*bn)
    1054             :   {
    1055        1016 :     *AB = cgetg(3, t_VEC);
    1056        1016 :     gel(*AB,1) = *A = cgetg(q+1, t_VEC);
    1057        1016 :     gel(*AB,2) = *B = cgetg(q+1, t_VEC);
    1058        1016 :     if (typ(an) == t_VEC) *an = RgV_kill0(*an);
    1059        1016 :     if (typ(bn) == t_VEC) *bn = RgV_kill0(*bn);
    1060             :   }
    1061             :   else
    1062             :   {
    1063       50787 :     *B = NULL;
    1064       50787 :     *AB = *A = cgetg(q+1, t_VEC);
    1065       50787 :     if (typ(*an) == t_VEC) *an = RgV_kill0(*an);
    1066             :   }
    1067       51803 : }
    1068             : GEN
    1069       14505 : lfuninit_theta2_worker(long r, GEN L, GEN qk, GEN a, GEN di, GEN an, GEN bn)
    1070             : {
    1071       14505 :   long q, m, prec = di[1], M = di[2], m0 = di[3], L0 = lg(an)-1;
    1072             :   GEN AB, A, B;
    1073       14505 :   worker_init((M - r) / m0 + 1, &an, &bn, &AB, &A, &B);
    1074      155888 :   for (q = 0, m = r; m <= M; m += m0, q++)
    1075             :   {
    1076      141381 :     GEN t = gel(qk, m+1);
    1077      141381 :     long N = minss(L[m+1],L0);
    1078      141380 :     gel(A, q+1) = theta2(an, N, t, a, prec); /* theta(exp(mh)) */
    1079      141381 :     if (bn) gel(B, q+1) = theta2(bn, N, t, a, prec);
    1080             :   }
    1081       14507 :   return AB;
    1082             : }
    1083             : 
    1084             : /* theta(exp(mh)) ~ sum_{n <= N} a(n) k[m,n] */
    1085             : static GEN
    1086      155087 : an_msum(GEN an, long N, GEN vKm)
    1087             : {
    1088      155087 :   pari_sp av = avma;
    1089      155087 :   GEN s = gen_0;
    1090             :   long n;
    1091    10053841 :   for (n = 1; n <= N; n++)
    1092     9899078 :     if (gel(vKm,n))
    1093             :     {
    1094     4425933 :       GEN c = mul_an(an, n, gel(vKm,n));
    1095     4425581 :       if (c) s = gadd(s, c);
    1096             :     }
    1097      154763 :   return gerepileupto(av, s);
    1098             : }
    1099             : 
    1100             : GEN
    1101       37297 : lfuninit_worker(long r, GEN K, GEN L, GEN peh2d, GEN vroots, GEN dr, GEN di,
    1102             :                 GEN an, GEN bn)
    1103             : {
    1104       37297 :   pari_sp av0 = avma;
    1105       37297 :   long m, n, q, L0 = lg(an)-1;
    1106       37297 :   double sig0 = rtodbl(gel(dr,1)), sub2 = rtodbl(gel(dr,2));
    1107       37297 :   double k1 = rtodbl(gel(dr,3)), MAXs = rtodbl(gel(dr,4));
    1108       37297 :   long D = di[1], M = di[2], m0 = di[3];
    1109       37297 :   double M0 = sig0? sub2 / sig0: 1./0.;
    1110       37297 :   GEN AB, A, B, vK = cgetg(M/m0 + 2, t_VEC);
    1111             : 
    1112      191409 :   for (q = 0, m = r; m <= M; m += m0, q++)
    1113      154113 :     gel(vK, q+1) = const_vec(L[m+1], NULL);
    1114       37296 :   worker_init(q, &an, &bn, &AB, &A, &B);
    1115      190742 :   for (m -= m0, q--; m >= 0; m -= m0, q--)
    1116             :   {
    1117      154111 :     double c1 = D + ((m > M0)? m * sig0 - sub2 : 0);
    1118      154111 :     GEN vKm = gel(vK,q+1); /* conceptually K(m,n) */
    1119    10052610 :     for (n = 1; n <= L[m+1]; n++)
    1120             :     {
    1121             :       GEN t2d, kmn;
    1122     9899164 :       long nn, mm, qq, p = 0;
    1123             :       double c, c2;
    1124             :       pari_sp av;
    1125             : 
    1126     9899164 :       if (gel(vKm, n)) continue; /* done already */
    1127     7652732 :       c = c1 + k1 * log2(n);
    1128             :       /* n *= 2; m -= m0 => c += c2, provided m >= M0. Else c += k1 */
    1129     7652732 :       c2 = k1 - MAXs;
    1130             :       /* p = largest (absolute) accuracy to which we need K(m,n) */
    1131    11672990 :       for (mm=m,nn=n; mm >= M0;)
    1132             :       {
    1133     8129482 :         if (nn <= L[mm+1] && (gel(an, nn) || (bn && gel(bn, nn))))
    1134     1489263 :           if (c > 0) p = maxuu(p, (ulong)c);
    1135     8129533 :         nn <<= 1;
    1136     8129533 :         mm -= m0; if (mm >= M0) c += c2; else { c += k1; break; }
    1137             :       }
    1138             :       /* mm < M0 || nn > L[mm+1] */
    1139    14746000 :       for (         ; mm >= 0; nn<<=1,mm-=m0,c+=k1)
    1140     7093239 :         if (nn <= L[mm+1] && (gel(an, nn) || (bn && gel(bn, nn))))
    1141     1709530 :           if (c > 0) p = maxuu(p, (ulong)c);
    1142     7652761 :       if (!p) continue; /* a_{n 2^v} = 0 for all v in range */
    1143     2179650 :       av = avma;
    1144     2179650 :       t2d = mpmul(gel(vroots,n), gel(peh2d,m+1));/*(n exp(mh)/sqrt(N))^(2/d)*/
    1145     2180224 :       kmn = gerepileupto(av, gammamellininvrt(K, t2d, p));
    1146     6733957 :       for (qq=q,mm=m,nn=n; mm >= 0; nn<<=1,mm-=m0,qq--)
    1147     4555001 :         if (nn <= L[mm+1]) gmael(vK, qq+1, nn) = kmn;
    1148             :     }
    1149             :   }
    1150      190701 :   for (q = 0, m = r; m <= M; m += m0, q++)
    1151             :   {
    1152      154074 :     long N = minss(L0, L[m+1]);
    1153      154074 :     gel(A, q+1) = an_msum(an, N, gel(vK,q+1));
    1154      154070 :     if (bn) gel(B, q+1) = an_msum(bn, N, gel(vK,q+1));
    1155             :   }
    1156       36627 :   return gerepileupto(av0, AB);
    1157             : }
    1158             : /* return A = [\theta(exp(mh)), m=0..M], theta(t) = sum a(n) K(n/sqrt(N) t),
    1159             :  * h = log(2)/m0. If bn != NULL, return the pair [A, B] */
    1160             : static GEN
    1161        5656 : lfuninit_ab(GEN theta, GEN h, struct lfunp *S)
    1162             : {
    1163        5656 :   const long M = S->M, prec = S->precmax;
    1164        5656 :   GEN tech = linit_get_tech(theta), isqN = theta_get_isqrtN(tech);
    1165        5656 :   GEN an = S->an, bn = S->bn, va, vb;
    1166             :   struct pari_mt pt;
    1167             :   GEN worker;
    1168             :   long m0, r, pending;
    1169             : 
    1170        5656 :   if (S->vgaell)
    1171             :   { /* d=2 and Vga = [a,a+1] */
    1172        2800 :     GEN a = vecmin(ldata_get_gammavec(linit_get_ldata(theta)));
    1173        2800 :     GEN qk = gpowers0(mpexp(h), M, isqN);
    1174        2800 :     m0 = minss(M+1, mt_nbthreads());
    1175        2800 :     worker = snm_closure(is_entry("_lfuninit_theta2_worker"),
    1176             :                          mkvecn(6, S->L, qk, a, mkvecsmall3(prec, M, m0),
    1177             :                                 an, bn? bn: gen_0));
    1178             :   }
    1179             :   else
    1180             :   {
    1181             :     GEN vroots, peh2d, d2;
    1182        2856 :     double sig0 = S->MAXs / S->m0, sub2 = S->sub / M_LN2;
    1183             :     /* For all 0<= m <= M, and all n <= L[m+1] such that a_n!=0, we compute
    1184             :      *   k[m,n] = K(n exp(mh)/sqrt(N))
    1185             :      * with ln(absolute error) <= E + max(mh sigma - sub, 0) + k1 * log(n).
    1186             :      * N.B. we use the 'rt' variant and pass (n exp(mh)/sqrt(N))^(2/d).
    1187             :      * Speedup: if n' = 2n and m' = m - m0 >= 0; then k[m,n] = k[m',n']. */
    1188        2856 :     vroots = mkvroots(S->d, S->nmax, prec); /* vroots[n] = n^(2/d) */
    1189        2856 :     d2 = gdivgu(gen_2, S->d);
    1190        2856 :     peh2d = gpowers0(gexp(gmul(d2,h), prec), M, gpow(isqN, d2, prec));
    1191        2856 :     m0 = S->m0; /* peh2d[m+1] = (exp(mh)/sqrt(N))^(2/d) */
    1192        2856 :     worker = snm_closure(is_entry("_lfuninit_worker"),
    1193             :                          mkvecn(8, theta_get_K(tech), S->L, peh2d, vroots,
    1194             :                                 mkvec4(dbltor(sig0), dbltor(sub2),
    1195             :                                        dbltor(S->k1), dbltor(S->MAXs)),
    1196             :                                 mkvecsmall3(S->D, M, m0),
    1197             :                                 an, bn? bn: gen_0));
    1198             :     /* For each 0 <= m <= M, we will sum for n<=L[m+1] a(n) K(m,n)
    1199             :      * bit accuracy for K(m,n): D + k1*log2(n) + 1_{m > M0} (m*sig0 - sub2)
    1200             :      * We restrict m to arithmetic progressions r mod m0 to save memory and
    1201             :      * allow parallelization */
    1202             :   }
    1203        5656 :   va = cgetg(M+2, t_VEC);
    1204        5656 :   vb = bn? cgetg(M+2, t_VEC): NULL;
    1205        5656 :   mt_queue_start_lim(&pt, worker, m0);
    1206        5656 :   pending = 0;
    1207       73976 :   for (r = 0; r < m0 || pending; r++)
    1208             :   { /* m = q m0 + r */
    1209             :     GEN done, A, B;
    1210             :     long q, m, workid;
    1211       68320 :     mt_queue_submit(&pt, r, r < m0 ? mkvec(utoi(r)): NULL);
    1212       68320 :     done = mt_queue_get(&pt, &workid, &pending);
    1213       68320 :     if (!done) continue;
    1214       51807 :     if (bn) { A = gel(done,1); B = gel(done,2); } else { A = done; B = NULL; }
    1215      347309 :     for (q = 0, m = workid; m <= M; m += m0, q++)
    1216             :     {
    1217      295502 :       gel(va, m+1) = gel(A, q+1);
    1218      295502 :       if (bn) gel(vb, m+1) = gel(B, q+1);
    1219             :     }
    1220             :   }
    1221        5656 :   mt_queue_end(&pt);
    1222        5656 :   return bn? mkvec2(va, vb): va;
    1223             : }
    1224             : 
    1225             : static void
    1226       93132 : parse_dom(double k, GEN dom, struct lfunp *S)
    1227             : {
    1228       93132 :   long l = lg(dom);
    1229       93132 :   if (typ(dom)!=t_VEC) pari_err_TYPE("lfuninit [domain]", dom);
    1230       93132 :   if (l == 2)
    1231             :   {
    1232       26125 :     S->dc = k/2.;
    1233       26125 :     S->dw = 0.;
    1234       26125 :     S->dh = gtodouble(gel(dom,1));
    1235             :   }
    1236       67007 :   else if (l == 3)
    1237             :   {
    1238         301 :     S->dc = k/2.;
    1239         301 :     S->dw = gtodouble(gel(dom,1));
    1240         301 :     S->dh = gtodouble(gel(dom,2));
    1241             :   }
    1242       66706 :   else if (l == 4)
    1243             :   {
    1244       66706 :     S->dc = gtodouble(gel(dom,1));
    1245       66706 :     S->dw = gtodouble(gel(dom,2));
    1246       66706 :     S->dh = gtodouble(gel(dom,3));
    1247             :   }
    1248             :   else
    1249             :   {
    1250           0 :     pari_err_TYPE("lfuninit [domain]", dom);
    1251           0 :     S->dc = S->dw = S->dh = 0; /*-Wall*/
    1252             :   }
    1253       93132 :   if (S->dw < 0 || S->dh < 0) pari_err_TYPE("lfuninit [domain]", dom);
    1254       93132 : }
    1255             : 
    1256             : /* do we have dom \subset dom0 ? dom = [center, width, height] */
    1257             : int
    1258       16084 : sdomain_isincl(double k, GEN dom, GEN dom0)
    1259             : {
    1260             :   struct lfunp S0, S;
    1261       16084 :   parse_dom(k, dom, &S);
    1262       16084 :   parse_dom(k, dom0, &S0);
    1263       16084 :   return S0.dc - S0.dw <= S.dc - S.dw
    1264       16084 :       && S0.dc + S0.dw >= S.dc + S.dw && S0.dh >= S.dh;
    1265             : }
    1266             : 
    1267             : static int
    1268       16084 : checklfuninit(GEN linit, GEN dom, long der, long bitprec)
    1269             : {
    1270       16084 :   GEN ldata = linit_get_ldata(linit);
    1271       16084 :   GEN domain = lfun_get_domain(linit_get_tech(linit));
    1272       16084 :   return domain_get_der(domain) >= der
    1273       16084 :     && domain_get_bitprec(domain) >= bitprec
    1274       32168 :     && sdomain_isincl(gtodouble(ldata_get_k(ldata)), dom, domain_get_dom(domain));
    1275             : }
    1276             : 
    1277             : static GEN
    1278        1043 : ginvsqrtvec(GEN x, long prec)
    1279             : {
    1280        1043 :   if (is_vec_t(typ(x)))
    1281        1064 :     pari_APPLY_same(ginv(gsqrt(gel(x,i), prec)))
    1282         742 :   else return ginv(gsqrt(x, prec));
    1283             : }
    1284             : 
    1285             : GEN
    1286        6489 : lfuninit_make(long t, GEN ldata, GEN tech, GEN domain)
    1287             : {
    1288        6489 :   GEN Vga = ldata_get_gammavec(ldata);
    1289        6489 :   long d = lg(Vga)-1;
    1290        6489 :   GEN w2 = gen_1, k2 = gmul2n(ldata_get_k(ldata), -1);
    1291        6489 :   GEN expot = gdivgu(gadd(gmulsg(d, gsubgs(k2, 1)), sumVga(Vga)), 4);
    1292        6489 :   if (typ(ldata_get_dual(ldata))==t_INT)
    1293             :   {
    1294        6335 :     GEN eno = ldata_get_rootno(ldata);
    1295        6335 :     long prec = nbits2prec( domain_get_bitprec(domain) );
    1296        6335 :     if (!isint1(eno)) w2 = ginvsqrtvec(eno, prec);
    1297             :   }
    1298        6489 :   tech = mkvec3(domain, tech, mkvec4(k2, w2, expot, gammafactor(Vga)));
    1299        6489 :   return mkvec3(mkvecsmall(t), ldata, tech);
    1300             : }
    1301             : 
    1302             : static void
    1303        2856 : lfunparams2(struct lfunp *S)
    1304             : {
    1305        2856 :   GEN L = S->L, an = S->an, bn = S->bn;
    1306             :   double pmax;
    1307        2856 :   long m, nan, nmax, neval, M = S->M;
    1308             : 
    1309        2856 :   S->vgaell = 0;
    1310             :   /* try to reduce parameters now we know the a_n (some may be 0) */
    1311        2856 :   if (typ(an) == t_VEC) an = RgV_kill0(an);
    1312        2856 :   nan = S->nmax; /* lg(an)-1 may be large than this */
    1313        2856 :   nmax = neval = 0;
    1314        2856 :   if (!bn)
    1315      155936 :     for (m = 0; m <= M; m++)
    1316             :     {
    1317      153101 :       long n = minss(nan, L[m+1]);
    1318      231025 :       while (n > 0 && !gel(an,n)) n--;
    1319      153101 :       if (n > nmax) nmax = n;
    1320      153101 :       neval += n;
    1321      153101 :       L[m+1] = n; /* reduce S->L[m+1] */
    1322             :     }
    1323             :   else
    1324             :   {
    1325          21 :     if (typ(bn) == t_VEC) bn = RgV_kill0(bn);
    1326        1036 :     for (m = 0; m <= M; m++)
    1327             :     {
    1328        1015 :       long n = minss(nan, L[m+1]);
    1329        1057 :       while (n > 0 && !gel(an,n) && !gel(bn,n)) n--;
    1330        1015 :       if (n > nmax) nmax = n;
    1331        1015 :       neval += n;
    1332        1015 :       L[m+1] = n; /* reduce S->L[m+1] */
    1333             :     }
    1334             :   }
    1335        2856 :   if (DEBUGLEVEL >= 1) err_printf("expected evaluations: %ld\n", neval);
    1336        2856 :   for (; M > 0; M--)
    1337        2856 :     if (L[M+1]) break;
    1338        2856 :   setlg(L, M+2);
    1339        2856 :   S->M = M;
    1340        2856 :   S->nmax = nmax;
    1341             : 
    1342             :   /* need K(n*exp(mh)/sqrt(N)) to absolute accuracy
    1343             :    *   D + k1*log(n) + max(m * sig0 - sub2, 0) */
    1344        2856 :   pmax = S->D + S->k1 * log2(L[1]);
    1345        2856 :   if (S->MAXs)
    1346             :   {
    1347        2856 :     double sig0 = S->MAXs/S->m0, sub2 = S->sub / M_LN2;
    1348      130428 :     for (m = ceil(sub2 / sig0); m <= S->M; m++)
    1349             :     {
    1350      127572 :       double c = S->D + m*sig0 - sub2;
    1351      127572 :       if (S->k1 > 0) c += S->k1 * log2(L[m+1]);
    1352      127572 :       pmax = maxdd(pmax, c);
    1353             :     }
    1354             :   }
    1355        2856 :   S->Dmax = pmax;
    1356        2856 :   S->precmax = nbits2prec(pmax);
    1357        2856 : }
    1358             : 
    1359             : static GEN
    1360        5670 : lfun_init_theta(GEN ldata, GEN eno, struct lfunp *S)
    1361             : {
    1362        5670 :   GEN an2, dual, tdom = NULL, Vga = ldata_get_gammavec(ldata);
    1363        5670 :   long L, prec = S->precmax;
    1364        5670 :   if (eno)
    1365        4004 :     L = S->nmax;
    1366             :   else
    1367             :   {
    1368        1666 :     tdom = dbltor(sqrt(0.5));
    1369        1666 :     L = maxss(S->nmax, lfunthetacost(ldata, tdom, 0, S->D));
    1370             :   }
    1371        5670 :   dual = ldata_get_dual(ldata);
    1372        5670 :   S->an = ldata_vecan(ldata_get_an(ldata), L, prec);
    1373        5656 :   S->bn = typ(dual)==t_INT? NULL: ldata_vecan(dual, S->nmax, prec);
    1374        5656 :   if (!vgaell(Vga)) lfunparams2(S);
    1375             :   else
    1376             :   {
    1377        2800 :     S->an = antwist(S->an, Vga, prec);
    1378        2800 :     if (S->bn) S->bn = antwist(S->bn, Vga, prec);
    1379        2800 :     S->vgaell = 1;
    1380             :   }
    1381        5656 :   an2 = lg(Vga)-1 == 1? antwist(S->an, Vga, prec): S->an;
    1382        5656 :   return lfunthetainit0(ldata, tdom, an2, 0, S->Dmax, 0);
    1383             : }
    1384             : 
    1385             : GEN
    1386       12098 : lfuncost(GEN L, GEN dom, long der, long bit)
    1387             : {
    1388       12098 :   pari_sp av = avma;
    1389       12098 :   GEN ldata = lfunmisc_to_ldata_shallow(L);
    1390       12098 :   GEN w, k = ldata_get_k(ldata);
    1391             :   struct lfunp S;
    1392             : 
    1393       12098 :   parse_dom(gtodouble(k), dom, &S);
    1394       12098 :   lfunp_set(ldata, der, bit, &S);
    1395       12098 :   w = ldata_get_rootno(ldata);
    1396       12098 :   if (isintzero(w)) /* for lfunrootres */
    1397           7 :     S.nmax = maxss(S.nmax, lfunthetacost(ldata, dbltor(sqrt(0.5)), 0, bit+1));
    1398       12098 :   set_avma(av); return mkvecsmall2(S.nmax, S.Dmax);
    1399             : }
    1400             : GEN
    1401          49 : lfuncost0(GEN L, GEN dom, long der, long bitprec)
    1402             : {
    1403          49 :   pari_sp av = avma;
    1404             :   GEN C;
    1405             : 
    1406          49 :   if (is_linit(L))
    1407             :   {
    1408          28 :     GEN tech = linit_get_tech(L);
    1409          28 :     GEN domain = lfun_get_domain(tech);
    1410          28 :     dom = domain_get_dom(domain);
    1411          28 :     der = domain_get_der(domain);
    1412          28 :     bitprec = domain_get_bitprec(domain);
    1413          28 :     if (linit_get_type(L) == t_LDESC_PRODUCT)
    1414             :     {
    1415          21 :       GEN v = lfunprod_get_fact(linit_get_tech(L)), F = gel(v,1);
    1416          21 :       long i, l = lg(F);
    1417          21 :       C = cgetg(l, t_VEC);
    1418          70 :       for (i = 1; i < l; ++i)
    1419          49 :         gel(C, i) = zv_to_ZV( lfuncost(gel(F,i), dom, der, bitprec) );
    1420          21 :       return gerepileupto(av, C);
    1421             :     }
    1422             :   }
    1423          28 :   if (!dom) pari_err_TYPE("lfuncost [missing s domain]", L);
    1424          28 :   C = lfuncost(L,dom,der,bitprec);
    1425          28 :   return gerepileupto(av, zv_to_ZV(C));
    1426             : }
    1427             : 
    1428             : GEN
    1429       22412 : lfuninit(GEN lmisc, GEN dom, long der, long bitprec)
    1430             : {
    1431       22412 :   pari_sp av = avma;
    1432             :   GEN poqk, AB, R, h, theta, ldata, eno, r, domain, tech, k;
    1433             :   struct lfunp S;
    1434             : 
    1435       22412 :   if (is_linit(lmisc))
    1436             :   {
    1437       16126 :     long t = linit_get_type(lmisc);
    1438       16126 :     if (t==t_LDESC_INIT || t==t_LDESC_PRODUCT)
    1439             :     {
    1440       16084 :       if (checklfuninit(lmisc, dom, der, bitprec)) return lmisc;
    1441          21 :       pari_warn(warner,"lfuninit: insufficient initialization");
    1442             :     }
    1443             :   }
    1444        6349 :   ldata = lfunmisc_to_ldata_shallow(lmisc);
    1445             : 
    1446        6349 :   switch (ldata_get_type(ldata))
    1447             :   {
    1448         595 :   case t_LFUN_NF:
    1449             :     {
    1450         595 :       GEN T = gel(ldata_get_an(ldata), 2);
    1451         595 :       return gerepilecopy(av, lfunzetakinit(T, dom, der, bitprec));
    1452             :     }
    1453          84 :   case t_LFUN_ABELREL:
    1454             :     {
    1455          84 :       GEN T = gel(ldata_get_an(ldata), 2);
    1456          84 :       return gerepilecopy(av, lfunabelianrelinit(gel(T,1), gel(T,2), dom, der, bitprec));
    1457             :     }
    1458             :   }
    1459        5670 :   k = ldata_get_k(ldata);
    1460        5670 :   parse_dom(gtodouble(k), dom, &S);
    1461        5670 :   lfunp_set(ldata, der, bitprec, &S);
    1462        5670 :   ldata = ldata_newprec(ldata, nbits2prec(S.Dmax));
    1463        5670 :   r = ldata_get_residue(ldata);
    1464             :   /* Note: all guesses should already have been performed (thetainit more
    1465             :    * expensive than needed: should be either tdom = 1 or bitprec = S.D).
    1466             :    * BUT if the root number / polar part do not have an algebraic
    1467             :    * expression, there is no way to do this until we know the
    1468             :    * precision, i.e. now. So we can't remove guessing code from here and
    1469             :    * lfun_init_theta */
    1470        5670 :   if (r && isintzero(r)) eno = NULL;
    1471             :   else
    1472             :   {
    1473        5670 :     eno = ldata_get_rootno(ldata);
    1474        5670 :     if (isintzero(eno)) eno = NULL;
    1475             :   }
    1476        5670 :   theta = lfun_init_theta(ldata, eno, &S);
    1477        5656 :   if (eno && !r)
    1478        2338 :     R = gen_0;
    1479             :   else
    1480             :   {
    1481        3318 :     GEN v = lfunrootres(theta, S.D);
    1482        3318 :     ldata = shallowcopy(ldata);
    1483        3318 :     gel(ldata, 6) = gel(v,3);
    1484        3318 :     r = gel(v,1);
    1485        3318 :     R = gel(v,2);
    1486        3318 :     if (isintzero(r)) setlg(ldata,7); else gel(ldata, 7) = r;
    1487             :   }
    1488        5656 :   h = divru(mplog2(S.precmax), S.m0);
    1489             :   /* exp(kh/2 . [0..M]) */
    1490        5656 :   poqk = gequal0(k) ? NULL
    1491        5656 :        : gpowers(gprec_w(mpexp(gmul2n(gmul(k,h), -1)), S.precmax), S.M);
    1492        5656 :   AB = lfuninit_ab(theta, h, &S);
    1493        5656 :   if (S.bn)
    1494             :   {
    1495         154 :     GEN A = gel(AB,1), B = gel(AB,2);
    1496         154 :     A = lfuninit_pol(A, poqk, S.precmax);
    1497         154 :     B = lfuninit_pol(B, poqk, S.precmax);
    1498         154 :     AB = mkvec2(A, B);
    1499             :   }
    1500             :   else
    1501        5502 :     AB = lfuninit_pol(AB, poqk, S.precmax);
    1502        5656 :   tech = mkvec3(h, AB, R);
    1503        5656 :   domain = mkvec2(dom, mkvecsmall2(der, bitprec));
    1504        5656 :   return gerepilecopy(av, lfuninit_make(t_LDESC_INIT, ldata, tech, domain));
    1505             : }
    1506             : 
    1507             : GEN
    1508         483 : lfuninit0(GEN lmisc, GEN dom, long der, long bitprec)
    1509             : {
    1510         483 :   GEN z = lfuninit(lmisc, dom, der, bitprec);
    1511         483 :   return z == lmisc? gcopy(z): z;
    1512             : }
    1513             : 
    1514             : /* If s is a pole of Lambda, return polar part at s; else return NULL */
    1515             : static GEN
    1516        5066 : lfunpoleresidue(GEN R, GEN s)
    1517             : {
    1518             :   long j;
    1519       14428 :   for (j = 1; j < lg(R); j++)
    1520             :   {
    1521        9915 :     GEN Rj = gel(R, j), be = gel(Rj, 1);
    1522        9915 :     if (gequal(s, be)) return gel(Rj, 2);
    1523             :   }
    1524        4513 :   return NULL;
    1525             : }
    1526             : 
    1527             : /* Compute contribution of polar part at s when not a pole. */
    1528             : static GEN
    1529        8221 : veccothderivn(GEN a, long n)
    1530             : {
    1531             :   long i;
    1532        8221 :   pari_sp av = avma;
    1533        8221 :   GEN c = pol_x(0), cp = mkpoln(3, gen_m1, gen_0, gen_1);
    1534        8221 :   GEN v = cgetg(n+2, t_VEC);
    1535        8221 :   gel(v, 1) = poleval(c, a);
    1536       24782 :   for(i = 2; i <= n+1; i++)
    1537             :   {
    1538       16561 :     c = ZX_mul(ZX_deriv(c), cp);
    1539       16561 :     gel(v, i) = gdiv(poleval(c, a), mpfact(i-1));
    1540             :   }
    1541        8221 :   return gerepilecopy(av, v);
    1542             : }
    1543             : 
    1544             : static GEN
    1545        8340 : polepart(long n, GEN h, GEN C)
    1546             : {
    1547        8340 :   GEN h2n = gpowgs(gdiv(h, gen_2), n-1);
    1548        8340 :   GEN res = gmul(h2n, gel(C,n));
    1549        8340 :   return odd(n)? res : gneg(res);
    1550             : }
    1551             : 
    1552             : static GEN
    1553        4079 : lfunsumcoth(GEN R, GEN s, GEN h, long prec)
    1554             : {
    1555             :   long i,j;
    1556        4079 :   GEN S = gen_0;
    1557       12300 :   for (j = 1; j < lg(R); ++j)
    1558             :   {
    1559        8221 :     GEN r = gel(R,j), be = gel(r,1), Rj = gel(r, 2);
    1560        8221 :     long e = valser(Rj);
    1561        8221 :     GEN z1 = gexpm1(gmul(h, gsub(s,be)), prec); /* exp(h(s-beta))-1 */
    1562        8221 :     GEN c1 = gaddgs(gdivsg(2, z1), 1); /* coth((h/2)(s-beta)) */
    1563        8221 :     GEN C1 = veccothderivn(c1, 1-e);
    1564       16561 :     for (i = e; i < 0; i++)
    1565             :     {
    1566        8340 :       GEN Rbe = mysercoeff(Rj, i);
    1567        8340 :       GEN p1 = polepart(-i, h, C1);
    1568        8340 :       S = gadd(S, gmul(Rbe, p1));
    1569             :     }
    1570             :   }
    1571        4079 :   return gmul2n(S, -1);
    1572             : }
    1573             : 
    1574             : static GEN lfunlambda_OK(GEN linit, GEN s, GEN sdom, long bitprec);
    1575             : /* L is a t_LDESC_PRODUCT Linit */
    1576             : static GEN
    1577        1426 : lfunlambda_product(GEN L, GEN s, GEN sdom, long bitprec)
    1578             : {
    1579        1426 :   GEN ldata = linit_get_ldata(L), v = lfunprod_get_fact(linit_get_tech(L));
    1580        1426 :   GEN r = gen_1, F = gel(v,1), E = gel(v,2), C = gel(v,3), cs = conj_i(s);
    1581        1426 :   long i, l = lg(F), isreal = gequal(imag_i(s), imag_i(cs));
    1582        4677 :   for (i = 1; i < l; ++i)
    1583             :   {
    1584        3251 :     GEN f = lfunlambda_OK(gel(F, i), s, sdom, bitprec);
    1585        3251 :     if( DEBUGLEVEL>=2) err_printf("lfunlambda(%ld): %Ps\n",i,f);
    1586        3251 :     if (typ(f)==t_VEC) f = RgV_prod(f);
    1587        3251 :     if (E[i]) r = gmul(r, gpowgs(f, E[i]));
    1588        3251 :     if (C[i])
    1589             :     {
    1590           0 :       GEN fc = isreal? f: conj_i(lfunlambda_OK(gel(F, i), cs, sdom, bitprec));
    1591           0 :       r = gmul(r, gpowgs(fc, C[i]));
    1592             :     }
    1593             :   }
    1594        1426 :   return (ldata_isreal(ldata) && gequal0(imag_i(s)))? real_i(r): r;
    1595             : }
    1596             : 
    1597             : /* s a t_SER */
    1598             : static long
    1599        1401 : der_level(GEN s)
    1600        1401 : { return signe(s)? lg(s)-3: valser(s)-1; }
    1601             : 
    1602             : /* s a t_SER; return coeff(s, X^0) */
    1603             : static GEN
    1604         385 : ser_coeff0(GEN s) { return simplify_shallow(polcoef_i(s, 0, -1)); }
    1605             : 
    1606             : static GEN
    1607        9338 : get_domain(GEN s, GEN *dom, long *der)
    1608             : {
    1609        9338 :   GEN sa = s;
    1610        9338 :   *der = 0;
    1611        9338 :   switch(typ(s))
    1612             :   {
    1613           7 :     case t_POL:
    1614           7 :     case t_RFRAC: s = toser_i(s);
    1615         385 :     case t_SER:
    1616         385 :       *der = der_level(s);
    1617         385 :       sa = ser_coeff0(s);
    1618             :   }
    1619        9338 :   *dom = mkvec3(real_i(sa), gen_0, gabs(imag_i(sa),DEFAULTPREC));
    1620        9338 :   return s;
    1621             : }
    1622             : /* assume s went through get_domain and s/bitprec belong to domain */
    1623             : static GEN
    1624       23024 : lfunlambda_OK(GEN linit, GEN s, GEN sdom, long bitprec)
    1625             : {
    1626             :   GEN eno, ldata, tech, h, pol;
    1627       23024 :   GEN S, S0 = NULL, k2, cost;
    1628             :   long prec, prec0;
    1629             :   struct lfunp D, D0;
    1630             : 
    1631       23024 :   if (linit_get_type(linit) == t_LDESC_PRODUCT)
    1632        1426 :     return lfunlambda_product(linit, s, sdom, bitprec);
    1633       21598 :   ldata = linit_get_ldata(linit);
    1634       21598 :   eno = ldata_get_rootno(ldata);
    1635       21598 :   tech = linit_get_tech(linit);
    1636       21598 :   h = lfun_get_step(tech); prec = realprec(h);
    1637             :   /* try to reduce accuracy */
    1638       21598 :   parse_dom(0, sdom, &D0);
    1639       21598 :   parse_dom(0, domain_get_dom(lfun_get_domain(tech)), &D);
    1640       21598 :   if (0.8 * D.dh > D0.dh)
    1641             :   {
    1642       12021 :     cost = lfuncost(linit, sdom, typ(s)==t_SER? der_level(s): 0, bitprec);
    1643       12021 :     prec0 = nbits2prec(cost[2]);
    1644       12021 :     if (prec0 < prec) { prec = prec0; h = gprec_w(h, prec); }
    1645             :   }
    1646       21598 :   pol = lfun_get_pol(tech);
    1647       21598 :   s = gprec_w(s, prec);
    1648       21598 :   if (ldata_get_residue(ldata))
    1649             :   {
    1650        4478 :     GEN R = lfun_get_Residue(tech);
    1651        4478 :     GEN Ra = lfunpoleresidue(R, s);
    1652        4478 :     if (Ra) return gprec_w(Ra, nbits2prec(bitprec));
    1653        4079 :     S0 = lfunsumcoth(R, s, h, prec);
    1654             :   }
    1655       21199 :   k2 = lfun_get_k2(tech);
    1656       21199 :   if (typ(pol)==t_POL && typ(s) != t_SER && gequal(real_i(s), k2))
    1657       15102 :   { /* on critical line: shortcut */
    1658       15102 :     GEN polz, b = imag_i(s);
    1659       15102 :     polz = gequal0(b)? poleval(pol,gen_1): poleval(pol, expIr(gmul(h,b)));
    1660       15102 :     S = gadd(polz, gmulvec(eno, conj_i(polz)));
    1661             :   }
    1662             :   else
    1663             :   {
    1664        6097 :     GEN z = gexp(gmul(h, gsub(s, k2)), prec);
    1665        6097 :     GEN zi = ginv(z), zc = conj_i(zi);
    1666        6097 :     if (typ(pol)==t_POL)
    1667        5901 :       S = gadd(poleval(pol, z), gmulvec(eno, conj_i(poleval(pol, zc))));
    1668             :     else
    1669         196 :       S = gadd(poleval(gel(pol,1), z), gmulvec(eno, poleval(gel(pol,2), zi)));
    1670             :   }
    1671       21199 :   if (S0) S = gadd(S,S0);
    1672       21199 :   return gprec_w(gmul(S,h), nbits2prec(bitprec));
    1673             : }
    1674             : GEN
    1675        1008 : lfunlambda(GEN lmisc, GEN s, long bitprec)
    1676             : {
    1677        1008 :   pari_sp av = avma;
    1678             :   GEN linit, dom, z;
    1679             :   long der;
    1680        1008 :   s = get_domain(s, &dom, &der);
    1681        1008 :   linit = lfuninit(lmisc, dom, der, bitprec);
    1682        1008 :   z = lfunlambda_OK(linit,s, dom, bitprec);
    1683        1008 :   return gerepilecopy(av, z);
    1684             : }
    1685             : 
    1686             : static long
    1687        8820 : is_ser(GEN x)
    1688             : {
    1689        8820 :   long t = typ(x);
    1690        8820 :   if (t == t_SER) return 1;
    1691        6874 :   if (!is_vec_t(t) || lg(x)==1) return 0;
    1692         350 :   if (typ(gel(x,1))==t_SER) return 1;
    1693         252 :   return 0;
    1694             : }
    1695             : 
    1696             : static GEN
    1697         357 : lfunser(GEN L)
    1698             : {
    1699         357 :   long v = valser(L);
    1700         357 :   if (v > 0) return gen_0;
    1701         315 :   if (v == 0) L = gel(L, 2);
    1702             :   else
    1703         189 :     setlg(L, minss(lg(L), 2-v));
    1704         315 :   return L;
    1705             : }
    1706             : 
    1707             : static GEN
    1708         357 : lfunservec(GEN x)
    1709             : {
    1710         357 :   if (typ(x)==t_SER) return lfunser(x);
    1711           0 :   pari_APPLY_same(lfunser(gel(x,i)))
    1712             : }
    1713             : static GEN
    1714         105 : lfununext(GEN L)
    1715             : {
    1716         105 :   setlg(L, maxss(lg(L)-1, valser(L)? 2: 3));
    1717         105 :   return normalizeser(L);
    1718             : }
    1719             : static GEN
    1720         105 : lfununextvec(GEN x)
    1721             : {
    1722         105 :   if (typ(x)==t_SER) return lfununext(x);
    1723           0 :   pari_APPLY_same(lfununext(gel(x,i)));
    1724             : }
    1725             : 
    1726             : /* assume lmisc is an linit, s went through get_domain and s/bitprec belong
    1727             :  * to domain */
    1728             : static GEN
    1729        5033 : lfun_OK(GEN linit, GEN s, GEN sdom, long bitprec)
    1730             : {
    1731        5033 :   GEN N, gas, S, FVga, res, ss = s;
    1732        5033 :   long prec = nbits2prec(bitprec), ext;
    1733             : 
    1734        5033 :   FVga = lfun_get_factgammavec(linit_get_tech(linit));
    1735        5033 :   S = lfunlambda_OK(linit, s, sdom, bitprec);
    1736        5033 :   if (is_ser(S))
    1737             :   {
    1738        1624 :     GEN r = typ(S)==t_SER ? S : gel(S,1);
    1739        1624 :     long d = lg(r) - 2 + fracgammadegree(FVga);
    1740        1624 :     if (typ(s) == t_SER)
    1741        1309 :       ss = sertoser(s, d);
    1742             :     else
    1743         315 :       ss = deg1ser_shallow(gen_1, s, varn(r), d);
    1744             :   }
    1745        5033 :   gas = gammafactproduct(FVga, ss, &ext, prec);
    1746        5033 :   N = ldata_get_conductor(linit_get_ldata(linit));
    1747        5033 :   res = gdiv(S, gmul(gpow(N, gdivgu(ss, 2), prec), gas));
    1748        5033 :   if (typ(s) != t_SER && is_ser(res)) res = lfunservec(res);
    1749        4676 :   else if (ext) res = lfununextvec(res);
    1750        5033 :   return gprec_w(res, prec);
    1751             : }
    1752             : 
    1753             : GEN
    1754        6762 : lfun(GEN lmisc, GEN s, long bitprec)
    1755             : {
    1756        6762 :   pari_sp av = avma;
    1757             :   GEN linit, dom, z;
    1758             :   long der;
    1759        6762 :   s = get_domain(s, &dom, &der);
    1760        6762 :   if (!der && typ(s) == t_INT && !is_bigint(s))
    1761             :   { /* special value ? */
    1762             :     GEN ldata;
    1763        5327 :     long t, ss = itos(s);
    1764        5327 :     if (is_linit(lmisc))
    1765         616 :       ldata = linit_get_ldata(lmisc);
    1766             :     else
    1767        4711 :       lmisc = ldata = lfunmisc_to_ldata_shallow(lmisc);
    1768        5320 :     t = ldata_get_type(ldata);
    1769        5320 :     if (t == t_LFUN_KRONECKER || t == t_LFUN_ZETA)
    1770             :     {
    1771        2744 :       long D = itos_or_0(gel(ldata_get_an(ldata), 2));
    1772        2744 :       if (D)
    1773             :       {
    1774        2744 :         if (ss <= 0) return lfunquadneg(D, ss);
    1775             :         /* ss > 0 */
    1776         238 :         if ((!odd(ss) && D > 0) || (odd(ss) && D < 0))
    1777             :         {
    1778         168 :           long prec = nbits2prec(bitprec), q = labs(D);
    1779         168 :           ss = 1 - ss; /* <= 0 */
    1780         168 :           z = powrs(divrs(mppi(prec + EXTRAPREC64), q), 1-ss);
    1781         168 :           z = mulrr(shiftr(z, -ss), sqrtr_abs(utor(q, prec)));
    1782         168 :           z = gdiv(z, mpfactr(-ss, prec));
    1783         168 :           if (smodss(ss, 4) > 1) togglesign(z);
    1784         168 :           return gmul(z, lfunquadneg(D, ss));
    1785             :         }
    1786             :       }
    1787             :     }
    1788             :   }
    1789        4081 :   linit = lfuninit(lmisc, dom, der, bitprec);
    1790        4067 :   z = lfun_OK(linit, s, dom, bitprec);
    1791        4067 :   return gerepilecopy(av, z);
    1792             : }
    1793             : 
    1794             : /* given a t_SER a+x*s(x), return x*s(x), shallow */
    1795             : static GEN
    1796          42 : sersplit1(GEN s, GEN *head)
    1797             : {
    1798          42 :   long i, l = lg(s);
    1799             :   GEN y;
    1800          42 :   *head = simplify_shallow(mysercoeff(s, 0));
    1801          42 :   if (valser(s) > 0) return s;
    1802          28 :   y = cgetg(l-1, t_SER); y[1] = s[1];
    1803          28 :   setvalser(y, 1);
    1804         140 :   for (i=3; i < l; i++) gel(y,i-1) = gel(s,i);
    1805          28 :   return normalizeser(y);
    1806             : }
    1807             : 
    1808             : /* order of pole of Lambda at s (0 if regular point) */
    1809             : static long
    1810        2198 : lfunlambdaord(GEN linit, GEN s)
    1811             : {
    1812        2198 :   GEN tech = linit_get_tech(linit);
    1813        2198 :   if (linit_get_type(linit)==t_LDESC_PRODUCT)
    1814             :   {
    1815         280 :     GEN v = lfunprod_get_fact(linit_get_tech(linit));
    1816         280 :     GEN F = gel(v, 1), E = gel(v, 2), C = gel(v, 3);
    1817         280 :     long i, ex = 0, l = lg(F);
    1818         952 :     for (i = 1; i < l; i++)
    1819         672 :       ex += lfunlambdaord(gel(F,i), s) * (E[i]+C[i]);
    1820         280 :     return ex;
    1821             :   }
    1822        1918 :   if (ldata_get_residue(linit_get_ldata(linit)))
    1823             :   {
    1824         588 :     GEN r = lfunpoleresidue(lfun_get_Residue(tech), s);
    1825         588 :     if (r) return lg(r)-2;
    1826             :   }
    1827        1764 :   return 0;
    1828             : }
    1829             : 
    1830             : static GEN
    1831         126 : derser(GEN res, long m)
    1832             : {
    1833         126 :   long v = valser(res);
    1834         126 :   if (v > m) return gen_0;
    1835         126 :   if (v >= 0)
    1836         126 :     return gmul(mysercoeff(res, m), mpfact(m));
    1837             :   else
    1838           0 :     return derivn(res, m, -1);
    1839             : }
    1840             : 
    1841             : static GEN
    1842         189 : derservec(GEN x, long m) { pari_APPLY_same(derser(gel(x,i),m)) }
    1843             : 
    1844             : /* derivative of order m > 0 of L (flag = 0) or Lambda (flag = 1) */
    1845             : static GEN
    1846        1575 : lfunderiv(GEN lmisc, long m, GEN s, long flag, long bitprec)
    1847             : {
    1848        1575 :   pari_sp ltop = avma;
    1849        1575 :   GEN res, S = NULL, linit, dom;
    1850        1575 :   long der, prec = nbits2prec(bitprec);
    1851        1575 :   if (m <= 0) pari_err_DOMAIN("lfun", "D", "<=", gen_0, stoi(m));
    1852        1568 :   s = get_domain(s, &dom, &der);
    1853        1568 :   linit = lfuninit(lmisc, dom, der + m, bitprec);
    1854        1568 :   if (typ(s) == t_SER)
    1855             :   {
    1856          42 :     long v, l = lg(s)-1;
    1857             :     GEN sh;
    1858          42 :     if (valser(s) < 0) pari_err_DOMAIN("lfun","valuation", "<", gen_0, s);
    1859          42 :     S = sersplit1(s, &sh);
    1860          42 :     v = valser(S);
    1861          42 :     s = deg1ser_shallow(gen_1, sh, varn(S), m + (l+v-1)/v);
    1862             :   }
    1863             :   else
    1864             :   {
    1865        1526 :     long ex = lfunlambdaord(linit, s);
    1866             :     /* HACK: pretend lfuninit was done to right accuracy */
    1867        1526 :     if (gequal0(s)) s = gen_0;
    1868        1526 :     s = deg1ser_shallow(gen_1, s, 0, m+1+ex);
    1869             :   }
    1870        1568 :   res = flag ? lfunlambda_OK(linit, s, dom, bitprec):
    1871         966 :                lfun_OK(linit, s, dom, bitprec);
    1872        1568 :   if (S)
    1873          42 :     res = gsubst(derivn(res, m, -1), varn(S), S);
    1874        1526 :   else if (typ(res)==t_SER)
    1875             :   {
    1876        1463 :     long v = valser(res);
    1877        1463 :     if (v > m) { set_avma(ltop); return gen_0; }
    1878        1456 :     if (v >= 0)
    1879        1330 :       res = gmul(mysercoeff(res, m), mpfact(m));
    1880             :     else
    1881         126 :       res = derivn(res, m, -1);
    1882             :   }
    1883          63 :   else if (is_ser(res))
    1884          63 :     res = derservec(res, m);
    1885        1561 :   return gerepilecopy(ltop, gprec_w(res, prec));
    1886             : }
    1887             : 
    1888             : GEN
    1889        1463 : lfunlambda0(GEN lmisc, GEN s, long der, long bitprec)
    1890             : {
    1891         602 :   return der? lfunderiv(lmisc, der, s, 1, bitprec)
    1892        2065 :             : lfunlambda(lmisc, s, bitprec);
    1893             : }
    1894             : 
    1895             : GEN
    1896        6545 : lfun0(GEN lmisc, GEN s, long der, long bitprec)
    1897             : {
    1898         973 :   return der? lfunderiv(lmisc, der, s, 0, bitprec)
    1899        7511 :             : lfun(lmisc, s, bitprec);
    1900             : }
    1901             : 
    1902             : GEN
    1903       13137 : lfunhardy(GEN lmisc, GEN t, long bitprec)
    1904             : {
    1905       13137 :   pari_sp ltop = avma;
    1906       13137 :   long prec = nbits2prec(bitprec), d;
    1907             :   GEN argz, z, linit, ldata, tech, dom, w2, k2, E, h, a, k;
    1908             : 
    1909       13137 :   switch(typ(t))
    1910             :   {
    1911       13130 :     case t_INT: case t_FRAC: case t_REAL: break;
    1912           7 :     default: pari_err_TYPE("lfunhardy",t);
    1913             :   }
    1914             : 
    1915       13130 :   ldata = lfunmisc_to_ldata_shallow(lmisc);
    1916       13130 :   if (!is_linit(lmisc)) lmisc = ldata;
    1917       13130 :   k = ldata_get_k(ldata);
    1918       13130 :   d = ldata_get_degree(ldata);
    1919       13130 :   dom = mkvec3(gmul2n(k, -1), gen_0, gabs(t,LOWDEFAULTPREC));
    1920       13130 :   linit = lfuninit(lmisc, dom, 0, bitprec);
    1921       13130 :   tech = linit_get_tech(linit);
    1922       13130 :   w2 = lfun_get_w2(tech);
    1923       13130 :   k2 = lfun_get_k2(tech);
    1924       13130 :   E = lfun_get_expot(tech); /* 4E = d(k2 - 1) + real(vecsum(Vga)) */
    1925       13130 :   z = mkcomplex(k2, t);
    1926             :   /* more accurate than garg: k/2 in Q */
    1927       13130 :   argz = gequal0(k2)? Pi2n(-1, prec): gatan(gdiv(t, k2), prec);
    1928       13130 :   prec = precision(argz);
    1929             :   /* prec may have increased: don't lose accuracy if |z|^2 is exact */
    1930       13130 :   a = gsub(gmulsg(d, gmul(t, gmul2n(argz,-1))),
    1931             :            gmul(E, glog(gnorm(z),prec)));
    1932       13130 :   h = lfunlambda_OK(linit, z, dom, bitprec);
    1933       13130 :   if (!isint1(w2) && typ(ldata_get_dual(ldata))==t_INT)
    1934       10773 :     h = mulrealvec(h, w2);
    1935       13130 :   if (typ(h) == t_COMPLEX && gexpo(imag_i(h)) < -(bitprec >> 1))
    1936        2308 :     h = real_i(h);
    1937       13130 :   return gerepileupto(ltop, gmul(h, gexp(a, prec)));
    1938             : }
    1939             : 
    1940             : /* L = log(t); return  \sum_{i = 0}^{v-1}  R[-i-1] L^i/i! */
    1941             : static GEN
    1942        1806 : theta_pole_contrib(GEN R, long v, GEN L)
    1943             : {
    1944        1806 :   GEN s = mysercoeff(R,-v);
    1945             :   long i;
    1946        1911 :   for (i = v-1; i >= 1; i--)
    1947         105 :     s = gadd(mysercoeff(R,-i), gdivgu(gmul(s,L), i));
    1948        1806 :   return s;
    1949             : }
    1950             : /* subtract successively rather than adding everything then subtracting.
    1951             :  * The polar part is "large" and suffers from cancellation: a little stabler
    1952             :  * this way */
    1953             : static GEN
    1954        4235 : theta_add_polar_part(GEN S, GEN R, GEN t, long prec)
    1955             : {
    1956        4235 :   GEN logt = NULL;
    1957        4235 :   long j, l = lg(R);
    1958        6041 :   for (j = 1; j < l; j++)
    1959             :   {
    1960        1806 :     GEN Rj = gel(R,j), b = gel(Rj,1), Rb = gel(Rj,2);
    1961        1806 :     long v = -valser(Rb);
    1962        1806 :     if (v > 1 && !logt) logt = glog(t, prec);
    1963        1806 :     S = gsub(S, gmul(theta_pole_contrib(Rb,v,logt), gpow(t,b,prec)));
    1964             :   }
    1965        4235 :   return S;
    1966             : }
    1967             : 
    1968             : static long
    1969        3528 : lfuncheckfeq_i(GEN theta, GEN thetad, GEN t0, GEN t0i, long bitprec)
    1970             : {
    1971        3528 :   GEN ldata = linit_get_ldata(theta);
    1972             :   GEN S0, S0i, w, eno;
    1973        3528 :   long prec = nbits2prec(bitprec);
    1974        3528 :   if (thetad)
    1975          70 :     S0 = lfuntheta(thetad, t0, 0, bitprec);
    1976             :   else
    1977        3458 :     S0 = conj_i(lfuntheta(theta, conj_i(t0), 0, bitprec));
    1978        3528 :   S0i = lfuntheta(theta, t0i, 0, bitprec);
    1979             : 
    1980        3528 :   eno = ldata_get_rootno(ldata);
    1981        3528 :   if (ldata_get_residue(ldata))
    1982             :   {
    1983         903 :     GEN R = theta_get_R(linit_get_tech(theta));
    1984         903 :     if (gequal0(R))
    1985             :     {
    1986             :       GEN v, r;
    1987         105 :       long t = ldata_get_type(ldata);
    1988         105 :       if (t == t_LFUN_NF || t == t_LFUN_ABELREL)
    1989             :       { /* inefficient since theta not needed; no need to optimize for this
    1990             :            (artificial) query [e.g. lfuncheckfeq(t_POL)] */
    1991          42 :         GEN L = lfuninit(ldata,zerovec(3),0,bitprec);
    1992          42 :         return lfuncheckfeq(L,t0,bitprec);
    1993             :       }
    1994          63 :       v = lfunrootres(theta, bitprec);
    1995          63 :       r = gel(v,1);
    1996          63 :       if (gequal0(eno)) eno = gel(v,3);
    1997          63 :       R = lfunrtoR_i(ldata, r, eno, nbits2prec(bitprec));
    1998             :     }
    1999         861 :     S0i = theta_add_polar_part(S0i, R, t0, prec);
    2000             :   }
    2001        3486 :   if (gequal0(S0i) || gequal0(S0)) pari_err_PREC("lfuncheckfeq");
    2002             : 
    2003        3486 :   w = gdivvec(S0i, gmul(S0, gpow(t0, ldata_get_k(ldata), prec)));
    2004             :   /* missing rootno: guess it */
    2005        3486 :   if (gequal0(eno)) eno = lfunrootno(theta, bitprec);
    2006        3486 :   w = gsubvec(w, eno);
    2007        3486 :   if (thetad) w = gdivvec(w, eno); /* |eno| may be large in non-dual case */
    2008        3486 :   return gexpo(w);
    2009             : }
    2010             : 
    2011             : /* Check whether the coefficients, conductor, weight, polar part and root
    2012             :  * number are compatible with the functional equation at t0 and 1/t0.
    2013             :  * Different from lfunrootres. */
    2014             : long
    2015        3647 : lfuncheckfeq(GEN lmisc, GEN t0, long bitprec)
    2016             : {
    2017             :   GEN ldata, theta, thetad, t0i;
    2018             :   pari_sp av;
    2019             : 
    2020        3647 :   if (is_linit(lmisc) && linit_get_type(lmisc)==t_LDESC_PRODUCT)
    2021             :   {
    2022         154 :     GEN v = lfunprod_get_fact(linit_get_tech(lmisc)), F = gel(v,1);
    2023         154 :     long i, b = -bitprec, l = lg(F);
    2024         518 :     for (i = 1; i < l; i++) b = maxss(b, lfuncheckfeq(gel(F,i), t0, bitprec));
    2025         154 :     return b;
    2026             :   }
    2027        3493 :   av = avma;
    2028        3493 :   if (!t0)
    2029             :   { /* ~Pi/3 + I/7, some random complex number */
    2030        3318 :     t0 = mkcomplex(uutoQ(355,339), uutoQ(1,7));
    2031        3318 :     t0i = ginv(t0);
    2032             :   }
    2033         175 :   else if (gcmpgs(gnorm(t0), 1) < 0) { t0i = t0; t0 = ginv(t0); }
    2034         119 :   else t0i = ginv(t0);
    2035             :   /* |t0| >= 1 */
    2036        3493 :   theta = lfunthetacheckinit(lmisc, t0i, 0, bitprec);
    2037        3486 :   ldata = linit_get_ldata(theta);
    2038        3486 :   thetad = theta_dual(theta, ldata_get_dual(ldata));
    2039        3486 :   return gc_long(av, lfuncheckfeq_i(theta, thetad, t0, t0i, bitprec));
    2040             : }
    2041             : 
    2042             : /*******************************************************************/
    2043             : /*       Compute root number and residues                          */
    2044             : /*******************************************************************/
    2045             : /* round root number to \pm 1 if close to integer. */
    2046             : static GEN
    2047        3724 : ropm1(GEN w, long prec)
    2048             : {
    2049             :   long e;
    2050             :   GEN r;
    2051        3724 :   if (typ(w) == t_INT) return w;
    2052        3332 :   r = grndtoi(w, &e);
    2053        3332 :   return (e < -prec2nbits(prec)/2)? r: w;
    2054             : }
    2055             : 
    2056             : /* theta for t=1/sqrt(2) and t2==2t simultaneously, saving 25% of the work.
    2057             :  * Assume correct initialization (no thetacheck) */
    2058             : static void
    2059         406 : lfunthetaspec(GEN linit, long bitprec, GEN *pv, GEN *pv2)
    2060             : {
    2061         406 :   pari_sp av = avma, av2;
    2062             :   GEN t, Vga, an, K, ldata, thetainit, v, v2, vroots;
    2063             :   long L, prec, n, d;
    2064             : 
    2065         406 :   ldata = linit_get_ldata(linit);
    2066         406 :   thetainit = linit_get_tech(linit);
    2067         406 :   prec = nbits2prec(bitprec);
    2068         406 :   Vga = ldata_get_gammavec(ldata); d = lg(Vga)-1;
    2069         406 :   if (Vgaeasytheta(Vga))
    2070             :   {
    2071         196 :     GEN v2 = sqrtr(real2n(1, nbits2prec(bitprec)));
    2072         196 :     GEN v = shiftr(v2,-1);
    2073         196 :     *pv = lfuntheta(linit, v,  0, bitprec);
    2074         196 :     *pv2= lfuntheta(linit, v2, 0, bitprec);
    2075         196 :     return;
    2076             :   }
    2077         210 :   an = RgV_kill0( theta_get_an(thetainit) );
    2078         210 :   L = lg(an)-1;
    2079             :   /* to compute theta(1/sqrt(2)) */
    2080         210 :   t = ginv(gsqrt(gmul2n(ldata_get_conductor(ldata), 1), prec));
    2081             :   /* t = 1/sqrt(2N) */
    2082             : 
    2083             :   /* From then on, the code is generic and could be used to compute
    2084             :    * theta(t) / theta(2t) without assuming t = 1/sqrt(2) */
    2085         210 :   K = theta_get_K(thetainit);
    2086         210 :   vroots = mkvroots(d, L, prec);
    2087         210 :   t = gpow(t, gdivgu(gen_2, d), prec); /* rt variant: t->t^(2/d) */
    2088             :   /* v = \sum_{n <= L, n odd} a_n K(nt) */
    2089     1815212 :   for (v = gen_0, n = 1; n <= L; n+=2)
    2090             :   {
    2091     1815002 :     GEN tn, Kn, a = gel(an, n);
    2092             : 
    2093     1815002 :     if (!a) continue;
    2094      113729 :     av2 = avma;
    2095      113729 :     tn = gmul(t, gel(vroots,n));
    2096      113729 :     Kn = gammamellininvrt(K, tn, bitprec);
    2097      113729 :     v = gerepileupto(av2, gadd(v, gmul(a,Kn)));
    2098             :   }
    2099             :   /* v += \sum_{n <= L, n even} a_n K(nt), v2 = \sum_{n <= L/2} a_n K(2n t) */
    2100     1815114 :   for (v2 = gen_0, n = 1; n <= L/2; n++)
    2101             :   {
    2102     1814904 :     GEN t2n, K2n, a = gel(an, n), a2 = gel(an,2*n);
    2103             : 
    2104     1814904 :     if (!a && !a2) continue;
    2105      120484 :     av2 = avma;
    2106      120484 :     t2n = gmul(t, gel(vroots,2*n));
    2107      120484 :     K2n = gerepileupto(av2, gammamellininvrt(K, t2n, bitprec));
    2108      120484 :     if (a) v2 = gadd(v2, gmul(a, K2n));
    2109      120484 :     if (a2) v = gadd(v,  gmul(a2,K2n));
    2110             :   }
    2111         210 :   *pv = v;
    2112         210 :   *pv2 = v2;
    2113         210 :   gerepileall(av, 2, pv,pv2);
    2114             : }
    2115             : 
    2116             : static GEN
    2117         399 : Rtor(GEN a, GEN R, GEN ldata, long prec)
    2118             : {
    2119         399 :   GEN FVga = gammafactor(ldata_get_gammavec(ldata));
    2120         399 :   GEN Na = gpow(ldata_get_conductor(ldata), gdivgu(a,2), prec);
    2121             :   long ext;
    2122         399 :   return gdiv(R, gmul(Na, gammafactproduct(FVga, a, &ext, prec)));
    2123             : }
    2124             : 
    2125             : /* v = theta~(t), vi = theta(1/t) */
    2126             : static GEN
    2127        3374 : get_eno(GEN R, GEN k, GEN t, GEN v, GEN vi, long vx, long bitprec, long force)
    2128             : {
    2129        3374 :   long prec = nbits2prec(bitprec);
    2130        3374 :   GEN a0, a1, S = deg1pol(gmul(gpow(t,k,prec), gneg(v)), vi, vx);
    2131             : 
    2132        3374 :   S = theta_add_polar_part(S, R, t, prec);
    2133        3374 :   if (typ(S) != t_POL || degpol(S) != 1) return NULL;
    2134        3374 :   a1 = gel(S,3); if (!force && gexpo(a1) < -bitprec/4) return NULL;
    2135        3325 :   a0 = gel(S,2);
    2136        3325 :   return gdivvec(a0, gneg(a1));
    2137             : 
    2138             : }
    2139             : /* Return w using theta(1/t) - w t^k \bar{theta}(t) = polar_part(t,w).
    2140             :  * The full Taylor development of L must be known */
    2141             : GEN
    2142        3325 : lfunrootno(GEN linit, long bitprec)
    2143             : {
    2144             :   GEN ldata, t, eno, v, vi, R, thetad;
    2145        3325 :   long c = 0, prec = nbits2prec(bitprec), vx = fetch_var();
    2146             :   GEN k;
    2147             :   pari_sp av;
    2148             : 
    2149             :   /* initialize for t > 1/sqrt(2) */
    2150        3325 :   linit = lfunthetacheckinit(linit, dbltor(sqrt(0.5)), 0, bitprec);
    2151        3325 :   ldata = linit_get_ldata(linit);
    2152        3325 :   k = ldata_get_k(ldata);
    2153        3339 :   R = ldata_get_residue(ldata)? lfunrtoR_eno(ldata, pol_x(vx), prec)
    2154        3325 :                               : cgetg(1, t_VEC);
    2155        3325 :   t = gen_1;
    2156        3325 :   v = lfuntheta(linit, t, 0, bitprec);
    2157        3325 :   thetad = theta_dual(linit, ldata_get_dual(ldata));
    2158        3325 :   vi = !thetad ? conj_i(v): lfuntheta(thetad, t, 0, bitprec);
    2159        3325 :   eno = get_eno(R,k,t,vi,v, vx, bitprec, 0);
    2160        3325 :   if (!eno && !thetad)
    2161             :   { /* t = sqrt(2), vi = theta(1/t), v = theta(t) */
    2162           7 :     lfunthetaspec(linit, bitprec, &vi, &v);
    2163           7 :     t = sqrtr(utor(2, prec));
    2164           7 :     eno = get_eno(R,k,t,conj_i(v),vi, vx, bitprec, 0);
    2165             :   }
    2166        3325 :   av = avma;
    2167        3367 :   while (!eno)
    2168             :   {
    2169          42 :     t = addsr(1, shiftr(utor(pari_rand(), prec), -2-BITS_IN_LONG));
    2170             :     /* t in [1,1.25[ */
    2171           0 :     v = thetad? lfuntheta(thetad, t, 0, bitprec)
    2172          42 :               : conj_i(lfuntheta(linit, t, 0, bitprec));
    2173          42 :     vi = lfuntheta(linit, ginv(t), 0, bitprec);
    2174          42 :     eno = get_eno(R,k,t,v,vi, vx, bitprec, c++ == 5);
    2175          42 :     set_avma(av);
    2176             :   }
    2177        3325 :   delete_var(); return ropm1(eno,prec);
    2178             : }
    2179             : 
    2180             : /* Find root number and/or residues when L-function coefficients and
    2181             :    conductor are known. For the moment at most a single residue allowed. */
    2182             : GEN
    2183        3745 : lfunrootres(GEN data, long bitprec)
    2184             : {
    2185        3745 :   pari_sp ltop = avma;
    2186             :   GEN k, w, r, R, a, b, e, v, v2, be, ldata, linit;
    2187             :   long prec;
    2188             : 
    2189        3745 :   ldata = lfunmisc_to_ldata_shallow(data);
    2190        3745 :   r = ldata_get_residue(ldata);
    2191        3745 :   k = ldata_get_k(ldata);
    2192        3745 :   w = ldata_get_rootno(ldata);
    2193        3745 :   if (r) r = normalize_simple_pole(r, k);
    2194        3745 :   if (!r || residues_known(r))
    2195             :   {
    2196        3346 :     if (isintzero(w)) w = lfunrootno(data, bitprec);
    2197        3346 :     if (!r)
    2198        1708 :       r = R = gen_0;
    2199             :     else
    2200        1638 :       R = lfunrtoR_eno(ldata, w, nbits2prec(bitprec));
    2201        3346 :     return gerepilecopy(ltop, mkvec3(r, R, w));
    2202             :   }
    2203         399 :   linit = lfunthetacheckinit(data, dbltor(sqrt(0.5)), 0, bitprec);
    2204         399 :   prec = nbits2prec(bitprec);
    2205         399 :   if (lg(r) > 2) pari_err_IMPL("multiple poles in lfunrootres");
    2206             :   /* Now residue unknown, and r = [[be,0]]. */
    2207         399 :   be = gmael(r, 1, 1);
    2208         399 :   if (ldata_isreal(ldata) && gequalm1(w))
    2209           0 :     R = lfuntheta(linit, gen_1, 0, bitprec);
    2210             :   else
    2211             :   {
    2212         399 :     GEN p2k = gpow(gen_2,k,prec);
    2213         399 :     lfunthetaspec(linit, bitprec, &v2, &v);
    2214         399 :     if (gequal(gmulsg(2, be), k)) pari_err_IMPL("pole at k/2 in lfunrootres");
    2215         399 :     if (gequal(be, k))
    2216             :     {
    2217         133 :       a = conj_i(gsub(gmul(p2k, v), v2));
    2218         133 :       b = subiu(p2k, 1);
    2219         133 :       e = gmul(gsqrt(p2k, prec), gsub(v2, v));
    2220             :     }
    2221             :     else
    2222             :     {
    2223         266 :       GEN tk2 = gsqrt(p2k, prec);
    2224         266 :       GEN tbe = gpow(gen_2, be, prec);
    2225         266 :       GEN tkbe = gpow(gen_2, gdivgu(gsub(k, be), 2), prec);
    2226         266 :       a = conj_i(gsub(gmul(tbe, v), v2));
    2227         266 :       b = gsub(gdiv(tbe, tkbe), tkbe);
    2228         266 :       e = gsub(gmul(gdiv(tbe, tk2), v2), gmul(tk2, v));
    2229             :     }
    2230         399 :     if (isintzero(w))
    2231             :     { /* Now residue unknown, r = [[be,0]], and w unknown. */
    2232           7 :       GEN t0  = mkfrac(utoi(11),utoi(10));
    2233           7 :       GEN th1 = lfuntheta(linit, t0,  0, bitprec);
    2234           7 :       GEN th2 = lfuntheta(linit, ginv(t0), 0, bitprec);
    2235           7 :       GEN tbe = gpow(t0, gmulsg(2, be), prec);
    2236           7 :       GEN tkbe = gpow(t0, gsub(k, be), prec);
    2237           7 :       GEN tk2 = gpow(t0, k, prec);
    2238           7 :       GEN c = conj_i(gsub(gmul(tbe, th1), th2));
    2239           7 :       GEN d = gsub(gdiv(tbe, tkbe), tkbe);
    2240           7 :       GEN f = gsub(gmul(gdiv(tbe, tk2), th2), gmul(tk2, th1));
    2241           7 :       GEN D = gsub(gmul(a, d), gmul(b, c));
    2242           7 :       w = gdiv(gsub(gmul(d, e), gmul(b, f)), D);
    2243             :     }
    2244         399 :     w = ropm1(w, prec);
    2245         399 :     R = gdiv(gsub(e, gmul(a, w)), b);
    2246             :   }
    2247         399 :   r = normalize_simple_pole(Rtor(be, R, ldata, prec), be);
    2248         399 :   R = lfunrtoR_i(ldata, r, w, prec);
    2249         399 :   return gerepilecopy(ltop, mkvec3(r, R, w));
    2250             : }
    2251             : 
    2252             : /*******************************************************************/
    2253             : /*                           Zeros                                 */
    2254             : /*******************************************************************/
    2255             : struct lhardyz_t {
    2256             :   long bitprec, prec;
    2257             :   GEN linit;
    2258             : };
    2259             : 
    2260             : static GEN
    2261       12507 : lfunhardyzeros(void *E, GEN t)
    2262             : {
    2263       12507 :   struct lhardyz_t *S = (struct lhardyz_t*)E;
    2264       12507 :   GEN z = gprec_wensure(lfunhardy(S->linit, t, S->bitprec), S->prec);
    2265       12507 :   return typ(z) == t_VEC ? RgV_prod(z): z;
    2266             : }
    2267             : 
    2268             : /* initialize for computation on critical line up to height h, zero
    2269             :  * of order <= m */
    2270             : static GEN
    2271         525 : lfuncenterinit(GEN lmisc, double h, long m, long bitprec)
    2272             : {
    2273         525 :   if (m < 0)
    2274             :   { /* choose a sensible default */
    2275         525 :     if (!is_linit(lmisc) || linit_get_type(lmisc) != t_LDESC_INIT) m = 4;
    2276             :     else
    2277             :     {
    2278         448 :       GEN domain = lfun_get_domain(linit_get_tech(lmisc));
    2279         448 :       m = domain_get_der(domain);
    2280             :     }
    2281             :   }
    2282         525 :   return lfuninit(lmisc, mkvec(dbltor(h)), m, bitprec);
    2283             : }
    2284             : 
    2285             : long
    2286         525 : lfunorderzero(GEN lmisc, long m, long bitprec)
    2287             : {
    2288         525 :   pari_sp ltop = avma;
    2289             :   GEN eno, ldata, linit, k2;
    2290             :   long G, c0, c, st;
    2291             : 
    2292         525 :   if (is_linit(lmisc) && linit_get_type(lmisc) == t_LDESC_PRODUCT)
    2293             :   {
    2294          77 :     GEN M = gmael(linit_get_tech(lmisc), 2,1);
    2295          77 :     long i, l = lg(M);
    2296         259 :     for (c=0, i=1; i < l; i++) c += lfunorderzero(gel(M,i), m, bitprec);
    2297          77 :     return c;
    2298             :   }
    2299         448 :   linit = lfuncenterinit(lmisc, 0, m, bitprec);
    2300         448 :   ldata = linit_get_ldata(linit);
    2301         448 :   eno = ldata_get_rootno(ldata);
    2302         448 :   k2 = gmul2n(ldata_get_k(ldata), -1);
    2303         448 :   G = -bitprec/2;
    2304         448 :   c0 = 0; st = 1;
    2305         448 :   if (typ(eno) == t_VEC)
    2306             :   {
    2307          42 :     long i, l = lg(eno), cnt = l-1, s = 0;
    2308          42 :     GEN v = zero_zv(l-1);
    2309          42 :     if (ldata_isreal(ldata)) st = 2;
    2310          84 :     for (c = c0; cnt; c += st)
    2311             :     {
    2312          42 :       GEN L = lfun0(linit, k2, c, bitprec);
    2313         154 :       for (i = 1; i < l; i++)
    2314             :       {
    2315         112 :         if (v[i]==0 && gexpo(gel(L,i)) > G)
    2316             :         {
    2317         112 :           v[i] = c; cnt--; s += c;
    2318             :         }
    2319             :       }
    2320             :     }
    2321          42 :     return gc_long(ltop,s);
    2322             :   }
    2323             :   else
    2324             :   {
    2325         406 :     if (ldata_isreal(ldata)) { st = 2; if (!gequal1(eno)) c0 = 1; }
    2326         434 :     for (c = c0;; c += st)
    2327         434 :       if (gexpo(lfun0(linit, k2, c, bitprec)) > G) return gc_long(ltop, c);
    2328             :   }
    2329             : }
    2330             : 
    2331             : /* assume T1 * T2 > 0, T1 <= T2 */
    2332             : static void
    2333          84 : lfunzeros_i(struct lhardyz_t *S, GEN *pw, long *ct, GEN T1, GEN T2, long d,
    2334             :             GEN cN, GEN pi2, GEN pi2div, long precinit, long prec)
    2335             : {
    2336          84 :   GEN T = T1, w = *pw;
    2337          84 :   long W = lg(w)-1, s = gsigne(lfunhardyzeros(S, T1));
    2338             :   for(;;)
    2339         392 :   {
    2340         476 :     pari_sp av = avma;
    2341             :     GEN D, T0, z;
    2342         476 :     D = gcmp(T, pi2) < 0? cN
    2343         476 :                         : gadd(cN, gmulsg(d, glog(gdiv(T, pi2), prec)));
    2344         476 :     D = gdiv(pi2div, D);
    2345             :     for(;;)
    2346        7721 :     {
    2347             :       long s0;
    2348        8197 :       T0 = T; T = gadd(T, D);
    2349        8197 :       if (gcmp(T, T2) >= 0) T = T2;
    2350        8197 :       s0 = gsigne(lfunhardyzeros(S, T));
    2351        8197 :       if (s0 != s) { s = s0; break; }
    2352        7805 :       if (T == T2) { setlg(w, *ct); *pw = w; return; }
    2353             :     }
    2354         392 :     z = zbrent(S, lfunhardyzeros, T0, T, prec); /* T <= T2 */
    2355         392 :     gerepileall(av, 2, &T, &z);
    2356         392 :     if (*ct > W) { W *= 2; w = vec_lengthen(w, W); }
    2357         392 :     if (typ(z) == t_REAL) z  = rtor(z, precinit);
    2358         392 :     gel(w, (*ct)++) = z;
    2359             :   }
    2360             :   setlg(w, *ct); *pw = w;
    2361             : }
    2362             : GEN
    2363          84 : lfunzeros(GEN ldata, GEN lim, long divz, long bitprec)
    2364             : {
    2365          84 :   pari_sp ltop = avma;
    2366             :   GEN linit, pi2, pi2div, cN, w, T, h1, h2;
    2367          84 :   long i, d, NEWD, c, ct, s1, s2, prec, prec0 = nbits2prec(bitprec);
    2368             :   double maxt;
    2369             :   struct lhardyz_t S;
    2370             : 
    2371          84 :   if (is_linit(ldata) && linit_get_type(ldata) == t_LDESC_PRODUCT)
    2372             :   {
    2373           0 :     GEN M = gmael(linit_get_tech(ldata), 2,1);
    2374           0 :     long l = lg(M);
    2375           0 :     w = cgetg(l, t_VEC);
    2376           0 :     for (i = 1; i < l; i++) gel(w,i) = lfunzeros(gel(M,i), lim, divz, bitprec);
    2377           0 :     return gerepileupto(ltop, vecsort0(shallowconcat1(w), NULL, 0));
    2378             :   }
    2379          84 :   if (typ(lim) == t_VEC)
    2380             :   {
    2381          49 :     if (lg(lim) != 3 || gcmp(gel(lim,1),gel(lim,2)) >= 0)
    2382           7 :       pari_err_TYPE("lfunzeros",lim);
    2383          42 :     h1 = gel(lim,1);
    2384          42 :     h2 = gel(lim,2);
    2385          42 :     maxt = maxdd(fabs(gtodouble(h1)), fabs(gtodouble(h2)));
    2386             :   }
    2387             :   else
    2388             :   {
    2389          35 :     if (gcmp(lim,gen_0) <= 0) pari_err_TYPE("lfunzeros",lim);
    2390          35 :     h1 = gen_0;
    2391          35 :     h2 = lim;
    2392          35 :     maxt = gtodouble(h2);
    2393             :   }
    2394          77 :   S.linit = linit = lfuncenterinit(ldata, maxt, -1, bitprec);
    2395          77 :   S.bitprec = bitprec;
    2396          77 :   S.prec = prec0;
    2397          77 :   ldata = linit_get_ldata(linit);
    2398          77 :   d = ldata_get_degree(ldata);
    2399             : 
    2400          77 :   NEWD = minss((long) ceil(bitprec + (M_PI/(4*M_LN2)) * d * maxt),
    2401             :                lfun_get_bitprec(linit_get_tech(linit)));
    2402          77 :   prec = nbits2prec(NEWD);
    2403          77 :   cN = gdiv(ldata_get_conductor(ldata), gpowgs(Pi2n(-1, prec), d));
    2404          77 :   cN = gexpo(cN) >= 0? gaddsg(d, gmulsg(2, glog(cN, prec))): utoi(d);
    2405          77 :   pi2 = Pi2n(1, prec);
    2406          77 :   pi2div = gdivgu(pi2, labs(divz));
    2407          77 :   s1 = gsigne(h1);
    2408          77 :   s2 = gsigne(h2);
    2409          77 :   w = cgetg(100+1, t_VEC); c = 1; ct = 0; T = NULL;
    2410          77 :   if (s1 <= 0 && s2 >= 0)
    2411             :   {
    2412          56 :     GEN r = ldata_get_residue(ldata);
    2413          56 :     if (!r || gequal0(r))
    2414             :     {
    2415          35 :       ct = lfunorderzero(linit, -1, bitprec);
    2416          35 :       if (ct) T = real2n(-prec2nbits(prec) / (2*ct), prec);
    2417             :     }
    2418             :   }
    2419          77 :   if (s1 <= 0)
    2420             :   {
    2421          63 :     if (s1 < 0)
    2422          21 :       lfunzeros_i(&S, &w, &c, h1, T? negr(T): h2,
    2423             :                   d, cN, pi2, pi2div, prec0, prec);
    2424          63 :     if (ct)
    2425             :     {
    2426          21 :       long n = lg(w)-1;
    2427          21 :       if (c + ct >= n) w = vec_lengthen(w, n + ct);
    2428          84 :       for (i = 1; i <= ct; i++) gel(w,c++) = gen_0;
    2429             :     }
    2430             :   }
    2431          77 :   if (s2 > 0 && (T || s1 >= 0))
    2432          63 :     lfunzeros_i(&S, &w, &c, T? T: h1, h2, d, cN, pi2, pi2div, prec0, prec);
    2433          77 :   return gerepilecopy(ltop, w);
    2434             : }
    2435             : 
    2436             : /*******************************************************************/
    2437             : /*       Guess conductor                                           */
    2438             : /*******************************************************************/
    2439             : struct huntcond_t {
    2440             :   GEN k;
    2441             :   GEN theta, thetad;
    2442             :   GEN *pM, *psqrtM, *pMd, *psqrtMd;
    2443             : };
    2444             : 
    2445             : static void
    2446       11962 : condset(struct huntcond_t *S, GEN M, long prec)
    2447             : {
    2448       11962 :   *(S->pM) = M;
    2449       11962 :   *(S->psqrtM) = gsqrt(ginv(M), prec);
    2450       11962 :   if (S->thetad != S->theta)
    2451             :   {
    2452           0 :     *(S->pMd) = *(S->pM);
    2453           0 :     *(S->psqrtMd) = *(S->psqrtM);
    2454             :   }
    2455       11962 : }
    2456             : 
    2457             : /* M should eventually converge to N, the conductor. L has no pole. */
    2458             : static GEN
    2459        6888 : wrap1(void *E, GEN M)
    2460             : {
    2461        6888 :   struct huntcond_t *S = (struct huntcond_t*)E;
    2462             :   GEN thetainit, tk, p1, p1inv;
    2463        6888 :   GEN t = mkfrac(stoi(11), stoi(10));
    2464             :   long prec, bitprec;
    2465             : 
    2466        6888 :   thetainit = linit_get_tech(S->theta);
    2467        6888 :   bitprec = theta_get_bitprec(thetainit);
    2468        6888 :   prec = nbits2prec(bitprec);
    2469        6888 :   condset(S, M, prec);
    2470        6888 :   tk = gpow(t, S->k, prec);
    2471        6888 :   p1 = lfuntheta(S->thetad, t, 0, bitprec);
    2472        6888 :   p1inv = lfuntheta(S->theta, ginv(t), 0, bitprec);
    2473        6888 :   return glog(gabs(gmul(tk, gdiv(p1, p1inv)), prec), prec);
    2474             : }
    2475             : 
    2476             : /* M should eventually converge to N, the conductor. L has a pole. */
    2477             : static GEN
    2478        5032 : wrap2(void *E, GEN M)
    2479             : {
    2480        5032 :   struct huntcond_t *S = (struct huntcond_t*)E;
    2481             :   GEN t1k, t2k, p1, p1inv, p2, p2inv, thetainit, R;
    2482        5032 :   GEN t1 = mkfrac(stoi(11), stoi(10)), t2 = mkfrac(stoi(13), stoi(11));
    2483             :   GEN t1be, t2be, t1bemk, t2bemk, t1kmbe, t2kmbe;
    2484             :   GEN F11, F12, F21, F22, P1, P2, res;
    2485             :   long prec, bitprec;
    2486        5032 :   GEN k = S->k;
    2487             : 
    2488        5032 :   thetainit = linit_get_tech(S->theta);
    2489        5032 :   bitprec = theta_get_bitprec(thetainit);
    2490        5032 :   prec = nbits2prec(bitprec);
    2491        5032 :   condset(S, M, prec);
    2492             : 
    2493        5032 :   p1 = lfuntheta(S->thetad, t1, 0, bitprec);
    2494        5032 :   p2 = lfuntheta(S->thetad, t2, 0, bitprec);
    2495        5032 :   p1inv = lfuntheta(S->theta, ginv(t1), 0, bitprec);
    2496        5032 :   p2inv = lfuntheta(S->theta, ginv(t2), 0, bitprec);
    2497        5032 :   t1k = gpow(t1, k, prec);
    2498        5032 :   t2k = gpow(t2, k, prec);
    2499        5032 :   R = theta_get_R(thetainit);
    2500        5032 :   if (typ(R) == t_VEC)
    2501             :   {
    2502           0 :     GEN be = gmael(R, 1, 1);
    2503           0 :     t1be = gpow(t1, be, prec); t1bemk = gdiv(gsqr(t1be), t1k);
    2504           0 :     t2be = gpow(t2, be, prec); t2bemk = gdiv(gsqr(t2be), t2k);
    2505           0 :     t1kmbe = gdiv(t1k, t1be);
    2506           0 :     t2kmbe = gdiv(t2k, t2be);
    2507             :   }
    2508             :   else
    2509             :   { /* be = k */
    2510        5032 :     t1be = t1k; t1bemk = t1k; t1kmbe = gen_1;
    2511        5032 :     t2be = t2k; t2bemk = t2k; t2kmbe = gen_1;
    2512             :   }
    2513        5032 :   F11 = conj_i(gsub(gmul(gsqr(t1be), p1), p1inv));
    2514        5032 :   F12 = conj_i(gsub(gmul(gsqr(t2be), p2), p2inv));
    2515        5032 :   F21 = gsub(gmul(t1k, p1), gmul(t1bemk, p1inv));
    2516        5032 :   F22 = gsub(gmul(t2k, p2), gmul(t2bemk, p2inv));
    2517        5032 :   P1 = gsub(gmul(t1bemk, t1be), t1kmbe);
    2518        5032 :   P2 = gsub(gmul(t2bemk, t2be), t2kmbe);
    2519        5032 :   res = gdiv(gsub(gmul(P2,F21), gmul(P1,F22)),
    2520             :              gsub(gmul(P2,F11), gmul(P1,F12)));
    2521        5032 :   return glog(gabs(res, prec), prec);
    2522             : }
    2523             : 
    2524             : /* If flag = 0 (default) return all conductors found as integers. If
    2525             : flag = 1, return the approximations, not the integers. If flag = 2,
    2526             : return all, even nonintegers. */
    2527             : 
    2528             : static GEN
    2529          84 : checkconductor(GEN v, long bit, long flag)
    2530             : {
    2531             :   GEN w;
    2532          84 :   long e, j, k, l = lg(v);
    2533          84 :   if (flag == 2) return v;
    2534          84 :   w = cgetg(l, t_VEC);
    2535         322 :   for (j = k = 1; j < l; j++)
    2536             :   {
    2537         238 :     GEN N = grndtoi(gel(v,j), &e);
    2538         238 :     if (e < -bit) gel(w,k++) = flag ? gel(v,j): N;
    2539             :   }
    2540          84 :   if (k == 2) return gel(w,1);
    2541           7 :   setlg(w,k); return w;
    2542             : }
    2543             : 
    2544             : static GEN
    2545          98 : parse_maxcond(GEN maxN)
    2546             : {
    2547             :   GEN M;
    2548          98 :   if (!maxN)
    2549          49 :     M = utoipos(10000);
    2550          49 :   else if (typ(maxN) == t_VEC)
    2551             :   {
    2552          14 :     if (!RgV_is_ZV(maxN)) pari_err_TYPE("lfunconductor",maxN);
    2553          14 :     return ZV_sort_shallow(maxN);
    2554             :   }
    2555             :   else
    2556          35 :     M = maxN;
    2557          84 :   return (typ(M) == t_INT)? addiu(M, 1): gceil(M);
    2558             : }
    2559             : 
    2560             : GEN
    2561          98 : lfunconductor(GEN data, GEN maxcond, long flag, long bitprec)
    2562             : {
    2563             :   struct huntcond_t S;
    2564          98 :   pari_sp av = avma;
    2565          98 :   GEN ldata = lfunmisc_to_ldata_shallow(data);
    2566          98 :   GEN ld, r, v, theta, thetad, M, tdom, t0 = NULL, t0i = NULL;
    2567             :   GEN (*eval)(void *, GEN);
    2568             :   long prec;
    2569          98 :   M = parse_maxcond(maxcond);
    2570          98 :   r = ldata_get_residue(ldata);
    2571          98 :   if (typ(M) == t_VEC) /* select in list */
    2572             :   {
    2573          14 :     if (lg(M) == 1) { set_avma(av); return cgetg(1,t_VEC); }
    2574           7 :     eval = NULL; tdom = dbltor(0.7);
    2575             :   }
    2576          84 :   else if (!r) { eval = wrap1; tdom = uutoQ(10,11); }
    2577             :   else
    2578             :   {
    2579          21 :     if (typ(r) == t_VEC && lg(r) > 2)
    2580           0 :       pari_err_IMPL("multiple poles in lfunconductor");
    2581          21 :     eval = wrap2; tdom = uutoQ(11,13);
    2582             :   }
    2583          91 :   if (eval) bitprec += bitprec/2;
    2584          91 :   prec = nbits2prec(bitprec);
    2585          91 :   ld = shallowcopy(ldata);
    2586          91 :   gel(ld, 5) = eval? M: veclast(M);
    2587          91 :   theta = lfunthetainit_i(ld, tdom, 0, bitprec);
    2588          91 :   thetad = theta_dual(theta, ldata_get_dual(ldata));
    2589          91 :   gel(theta,3) = shallowcopy(linit_get_tech(theta));
    2590          91 :   S.k = ldata_get_k(ldata);
    2591          91 :   S.theta = theta;
    2592          91 :   S.thetad = thetad? thetad: theta;
    2593          91 :   S.pM = &gel(linit_get_ldata(theta),5);
    2594          91 :   S.psqrtM = &gel(linit_get_tech(theta),7);
    2595          91 :   if (thetad)
    2596             :   {
    2597           0 :     S.pMd = &gel(linit_get_ldata(thetad),5);
    2598           0 :     S.psqrtMd = &gel(linit_get_tech(thetad),7);
    2599             :   }
    2600          91 :   if (!eval)
    2601             :   {
    2602           7 :     long i, besti = 0, beste = -10, l = lg(M);
    2603           7 :     t0 = uutoQ(11,10); t0i = uutoQ(10,11);
    2604          49 :     for (i = 1; i < l; i++)
    2605             :     {
    2606          42 :       pari_sp av2 = avma;
    2607             :       long e;
    2608          42 :       condset(&S, gel(M,i), prec);
    2609          42 :       e = lfuncheckfeq_i(theta, thetad, t0, t0i, bitprec);
    2610          42 :       set_avma(av2);
    2611          42 :       if (e < beste) { beste = e; besti = i; }
    2612          35 :       else if (e == beste) beste = besti = 0; /* tie: forget */
    2613             :     }
    2614           7 :     if (!besti) { set_avma(av); return cgetg(1,t_VEC); }
    2615           7 :     return gerepilecopy(av, mkvec2(gel(M,besti), stoi(beste)));
    2616             :   }
    2617          84 :   v = solvestep((void*)&S, eval, ghalf, M, gen_2, 14, prec);
    2618          84 :   return gerepilecopy(av, checkconductor(v, bitprec/2, flag));
    2619             : }
    2620             : 
    2621             : /* assume chi primitive */
    2622             : static GEN
    2623         560 : znchargauss_i(GEN G, GEN chi, long bitprec)
    2624             : {
    2625         560 :   GEN z, q, F = znstar_get_N(G);
    2626             :   long prec;
    2627             : 
    2628         560 :   if (equali1(F)) return gen_1;
    2629         560 :   prec = nbits2prec(bitprec);
    2630         560 :   q = sqrtr_abs(itor(F, prec));
    2631         560 :   z = lfuntheta(mkvec2(G,chi), gen_1, 0, bitprec);
    2632         560 :   if (gexpo(z) < 10 - bitprec)
    2633             :   {
    2634          28 :     if (equaliu(F,300))
    2635             :     {
    2636          14 :       GEN z = rootsof1u_cx(25, prec);
    2637          14 :       GEN n = znconreyexp(G, chi);
    2638          14 :       if (equaliu(n, 131)) return gmul(q, gpowgs(z,14));
    2639           7 :       if (equaliu(n, 71)) return gmul(q, gpowgs(z,11));
    2640             :     }
    2641          14 :     if (equaliu(F,600))
    2642             :     {
    2643          14 :       GEN z = rootsof1u_cx(25, prec);
    2644          14 :       GEN n = znconreyexp(G, chi);
    2645          14 :       if (equaliu(n, 491)) return gmul(q, gpowgs(z,7));
    2646           7 :       if (equaliu(n, 11)) return gmul(q, gpowgs(z,18));
    2647             :     }
    2648           0 :     pari_err_BUG("znchargauss [ Theta(chi,1) = 0 ]");
    2649             :   }
    2650         532 :   z = gmul(gdiv(z, conj_i(z)), q);
    2651         532 :   if (zncharisodd(G,chi)) z = mulcxI(z);
    2652         532 :   return z;
    2653             : }
    2654             : static GEN
    2655         560 : Z_radical(GEN N, long *om)
    2656             : {
    2657         560 :   GEN P = gel(Z_factor(N), 1);
    2658         560 :   *om = lg(P)-1; return ZV_prod(P);
    2659             : }
    2660             : GEN
    2661        1218 : znchargauss(GEN G, GEN chi, GEN a, long bitprec)
    2662             : {
    2663             :   GEN v, T, N, F, b0, b1, b2, bF, a1, aF, A, r, GF, tau, B, faB, u, S;
    2664        1218 :   long omb0, prec = nbits2prec(bitprec);
    2665        1218 :   pari_sp av = avma;
    2666             : 
    2667        1218 :   if (typ(chi) != t_COL) chi = znconreylog(G,chi);
    2668        1218 :   T = znchartoprimitive(G, chi);
    2669        1218 :   GF  = gel(T,1);
    2670        1218 :   chi = gel(T,2); /* now primitive */
    2671        1218 :   N = znstar_get_N(G);
    2672        1218 :   F = znstar_get_N(GF);
    2673        1218 :   if (equalii(N,F)) b1 = bF = gen_1;
    2674             :   else
    2675             :   {
    2676         245 :     v = Z_ppio(diviiexact(N,F), F);
    2677         245 :     bF = gel(v,2); /* (N/F, F^oo) */
    2678         245 :     b1 = gel(v,3); /* cofactor */
    2679             :   }
    2680        1218 :   if (!a) a = a1 = aF = gen_1;
    2681             :   else
    2682             :   {
    2683        1169 :     if (typ(a) != t_INT) pari_err_TYPE("znchargauss",a);
    2684        1169 :     a = modii(a, N);
    2685        1169 :     if (!signe(a)) { set_avma(av); return is_pm1(F)? eulerphi(N): gen_0; }
    2686         728 :     v = Z_ppio(a, F);
    2687         728 :     aF = gel(v,2);
    2688         728 :     a1 = gel(v,3);
    2689             :   }
    2690         777 :   if (!equalii(aF, bF)) { set_avma(av); return gen_0; }
    2691         560 :   b0 = Z_radical(b1, &omb0);
    2692         560 :   b2 = diviiexact(b1, b0);
    2693         560 :   A = dvmdii(a1, b2, &r);
    2694         560 :   if (r != gen_0) { set_avma(av); return gen_0; }
    2695         560 :   B = gcdii(A,b0); faB = Z_factor(B); /* squarefree */
    2696         560 :   S = eulerphi(mkvec2(B,faB));
    2697         560 :   if (odd(omb0 + lg(gel(faB,1))-1)) S = negi(S); /* moebius(b0/B) * phi(B) */
    2698         560 :   S = mulii(S, mulii(aF,b2));
    2699         560 :   tau = znchargauss_i(GF, chi, bitprec);
    2700         560 :   u = Fp_div(b0, A, F);
    2701         560 :   if (!equali1(u))
    2702             :   {
    2703           7 :     GEN ord = zncharorder(GF, chi), z = rootsof1_cx(ord, prec);
    2704           7 :     tau = gmul(tau, znchareval(GF, chi, u, mkvec2(z,ord)));
    2705             :   }
    2706         560 :   return gerepileupto(av, gmul(tau, S));
    2707             : }

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