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Let $f$ be a primitive form of an even weight $k$ for the full modular group and let $L(Sym^rf,s)$ be the symmetric $r$th $(r\geq 2)$ power $L$-function associated to $f.$ I have three questions related to the function $L(Sym^rf,s):$

1) Is the function $L(Sym^rf,s)$ holomorphic on the whole real line?

2) Has it infinitely many real zeros?

3) What is its domain of convergence?

Many thanks, Khadija

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  • $\begingroup$ #1 I don't think it is proper to talk about "holomorphic on the whole real line", don't you need a domain for this? #2 There should be zeros at the poles of the $\Gamma$-factors, and so there should be infinitely many real zeros. #3 Do you mean the domain of (absolute?) convergence of the Dirichlet series? In any case, for #1 I think it is known that the second (Gelbart-Jacquet), third (Shahidi), and fourth (Kim-Shahidi) symmetric powers are functorial and are thus holomorphic (except possibly a pole), while the 5th-9th are meromorphic and satisfy a functional equation. $\endgroup$
    – kantelope
    Commented Sep 3, 2015 at 0:20
  • $\begingroup$ The Dirichlet series converges absolutely for $\sigma\gt 1+(k-1)r/2$ in the arithmetic normalization with $|c_r(p)|\le (r+1)p^{(k-1)r/2}$, and of course it's just $\sigma\gt 1$ if you renormalize to make all the coefficients $|c_r(p)|\le r+1$. If you are asking about Maass forms, the answer might be different for all I know. $\endgroup$
    – kantelope
    Commented Sep 3, 2015 at 0:24
  • $\begingroup$ Thank you @kantelope for your answers. For the notion of "holomorphic on the whole real line" I mean holomorphy on each real number so on R. For the other two answers, they are what I want to know! $\endgroup$
    – Khadija Mbarki
    Commented Sep 3, 2015 at 7:35

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