Xavier Roblot on Thu, 22 Oct 2015 18:03:59 +0200

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Re: GP interface for computing Artin L functions

> On 22 Oct 2015, at 14:50, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote:
> On Thu, Oct 22, 2015 at 02:18:41PM +0200, Xavier Roblot wrote:
>> Hi Bill, 
>>> We just added to master a new function lfunartin() to compute
>>> Artin L functions.
>>> This is based on a GP script by Charlotte Euvrard.
>>> Currently, the representation needs to be given explicitly
>>> This is the documentation:
>>> lfunartin(nf,gal,M,n):
>>>  Returns the Ldata structure associated to the Artin L-function associated to the
>>> representation  rho of the Galois group of the extension K/Q,  defined over the cyclotomic field
>>> Q(zeta_n),   where  nf is the nfinit structure associated to K,  gal is the galoisinit structure
>>> associated to K/Q, and M is the vector of the image of the generators G.gen by rho. The elements
>>> of  M  are matrices with polynomial entries,  whose variable is understood as the complex number
>>> exp(2 i Pi/n).
>> This is great news and that will be very useful! Do you think it could be
>> possible to also specify a finite set of prime ideals of K (maybe by
>> providing an integral ideal of K) that should be excluded from the Euler
>> product defining the L-function? This kind of L-functions are very often used
>> in the context of Stark conjectures. 
> Does such L functions satisfy a functional equation compatible with the lfun
> interface ? 

Well, I am not sure about that. But, usually, the way to do it is to compute the value of the primitive L-function and then multiply it by the right factor that is relatively easy to compute. In fact, I think I could even probably write the corresponding code if I have your blessing ;)