Re: L-functions of ray class fields

[Markus Grassl <markus.grassl@ug.edu.pl>]

Merci!

I should be able to adapt this to other examples.


Markus

Am 13/09/2021 um 17:08 schrieb Bill Allombert:
> 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