Re: L-functions twisted by character

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

On Sat, Sep 17, 2016 at 10:16:02AM +0200, Bill Allombert wrote:
> On Fri, Sep 16, 2016 at 11:14:09AM +0200, Jeroen Demeyer wrote:
> > Does PARI support L-functions (in particular of elliptic curves) twisted by
> > a character? I couldn't find it explicitly in the manual, but I might have
> > missed something.
> 
> No it is not implemented. The problem is to compute correctly the
> conductor and the Euler factors at the bad primes.

I added a function lfuntwist which allows to twist by an arbitrary
primitive Dirichlet character, however it requires the conductor of
the character to be prime to the conductor of the L-function, which
avoid the above issue.

In this example, we twist the L function of an elliptic curve by a
character of order 3 and of conductor 7.

? E=ellinit("11a1");
? nf=nfinit(polsubcyclo(7,3,'a));
? E2=ellinit(E[1..5],nf);
? L=lfuntwist(E,Mod(2,7));
? lfun(E,2)*norm(lfun(L,2))-lfun(E2,2)
%24 = -7.560250692031083222E-34

Due to the restriction, for L function of elliptic curves twisted by a
quadratic character it is better to twist the curve instead.

Cheers,
Bill.