Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hello,

I would like to know whether, given an algebraic number $\alpha$ of degree $d$, the Dedekind Zeta function $\zeta_{\mathbb{Q}(\alpha)}$ is always a function of the Selberg class of degree $d$ of not. I know that it is true when $\mathbb{Q}(\alpha)$ is an abelian extension of $\mathbb{Q}$, but what about the non abelian case? Thank you in advance.

share|improve this question
add comment

1 Answer 1

up vote 5 down vote accepted

Yes, even the broader class of Hecke L-functions (general L-functions for GL(1) over number fields) are in Selberg's class, for straightforward reasons coming from the functional equation proven by Hecke (and redone by Iwasawa-Tate).

Perhaps the least obvious part is the "order" requirement, but this is what follows from the functional equation and from the Laplace-Stirling asymptotics for the Gamma function.

share|improve this answer
    
Thank you very much for this fast and nice answer. –  Sylvain JULIEN Jun 20 '11 at 18:52
    
Perhaps it is worthwhile to remark that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}(n)$ over a number field with unitary central character, and $\pi$ satisfies the Ramanujan conjecture, then $L(s,\pi)$ is in the Selberg class. Moreover, it is expected that any element in the Selberg class is of that form. –  GH from MO Jun 22 '11 at 4:02
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.