Re: bnrL1 output ordering

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Fri, Jan 06, 2023 at 09:11:29PM +0100, Pierre Charollois wrote:
> Thanks Bill !
> 
> I am indeed interested in Dirichlet series that are the sum over ideals in
> a given ideal class (including a sign character) for complex cubic fields.

What kind of sign character ?

You can try this one that take a bnr as input, so you can include
places at infinity.

charall(c)=
{
  my(L=List());
  forvec(v=vector(#c,i,[0,c[i]-1]),listput(L,v));
  Vec(L);
}

partialzeta(B,s)=
{
  my(pr=idealfactor(B,B.mod)[,1]);
  my(C=charall(B.cyc), d=lcm(B.cyc), M=matdiagonal(B.cyc)^-1);
  my(V=apply(c->lfun([B,c],s),C));
  my(W=apply(c->prod(i=1,#pr,1-pr[i].p^(-s*pr[i].f)*chareval(B,c,pr[i])),C));
  vector(#C,j, sum(i=1,#V,exp(2*I*Pi*C[i]*M*C[j]~)*V[i]*W[i]))/#C
}

? bnf=bnfinit(a^3-37);bnr=bnrinit(bnf,[1,[1]]);
? partialzeta(bnr,2)
%2 = [1.1364431204298725596424145099053270595+0.E-40*I,0.43276840644100776464031947320118672180+9.795786256852395900E-40*I,0.36327588248510712879791291401014770222+0.E-38*I]
? vecsum(%)
%3 = 1.9324874093559874530806468971166614835+0.E-38*I
? lfun(bnr,2)
%4 = 1.9324874093559874530806468971166614835

Cheers,
Bill