Re: L-functions of ray class fields

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

On Mon, Sep 13, 2021 at 03:46:42PM +0200, Bill Allombert wrote:
> > 
> > But this does not remove the Euler factors, which I request above by setting
> > the third bit of the flag.
> > Is there a simple way to define the L-function with the Euler factors
> > removed so that I can get particular values as they would be returned by
> > bnrL1?
> 
> No, because such imprimitive functions do not satisfy the functional
> equation. But you can multiply by the Euler factors afterward.

Computing the Euler factor is a bit technical:

K=bnfinit(y^2-17);
R=bnrinit(K,[idealmul(K,2,idealprimedec(K,67)[1]),[1,0]]);
\\ Let chi be your character:
chi=[1]
\\ compute the conductor of chi and the corresponding ray class group:
Rchi=bnrinit(R,bnrconductor(R,chi));
\\ compute the set of place S thta divide the modulus of R
S=idealfactor(R,R.mod[1])[,1];
\\define the Euler factor:
E(s)=
{
  my(o=Rchi.cyc[1],z=[exp(2*I*Pi/o),o]);
  prod(i=1,#F,1/(1-chareval(Rchi,chi,S[i],z)*idealnorm(Rchi,S[i])^-s))
}

\\ check
E(1)
bnrL1(R,,1)[1]/bnrL1(R,,3)[chi[1]]
E(0)
bnrL1(R,,0)[1][2]/bnrL1(R,,2)[chi[1]][2]

Cheers,
Bill