Re: L-functions of ray class fields

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

On Mon, Sep 13, 2021 at 03:13:21PM +0200, Markus Grassl wrote:
> Hello,
> 
> I am starting to explore the functionality to evaluate L-functions of ray
> class fields in PARI.
> 
> Consider the following example:
> 
> ? K=bnfinit(y^2-17);
> ? R=bnrinit(K,[idealmul(K,2,idealprimedec(K,67)[1]),[1,0]]);
> ? bnrL1(R,,6)
...

> First, it seems as if the function bnrL1 was single-threaded, while lfun
> used multiple cores. 

Indeed this is the case.

> Is there a simple way to use multi-threading?

Could you clarify ?

Do you mean, to have bnrL1 use multithreading ? it is not done yet.
On the other hand you can use lfun with a vector of characters if it is
that you had in mind.

> I tried to directly evaluate the L-functions, e.g.
> 
> ? lfun(lfuncreate([R,[1]]),0,1)
> %7 = 0.90384905518988545678200390170972794465 -
> 2.3724351853612471172697035833485504030*I
> 
> 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.

Cheers,
Bill