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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.

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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.

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  • $\begingroup$ 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. $\endgroup$
    – GH from MO
    Jun 22, 2011 at 4:02

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