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.



Algebraic numbers and Selberg class
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.


