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Let $L(s,\pi)$ be the standard(Godement-Jacquet) $L$-function of $\pi$, where $\pi$ is a cuspidal automorphic represetation of $GL(m,A_Q)$.

What's the standard zero-free region for $L(s,\pi)$? any reference?

ANy progress beyond standard one?

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    $\begingroup$ There are some results about zero-free regions of L-functions in section 5.4 of Iwaniec and Kowalski's book on analytic number theory, though there are some technical conditions on their results and I don't know if they are known to hold in your case. Certainly studying free-regions of L-functions is a big business. $\endgroup$ Jun 3, 2014 at 14:02
  • $\begingroup$ If you like my answer, please accept it officially (so that it turns green). Thanks in advance! $\endgroup$
    – GH from MO
    Aug 19, 2018 at 23:13

1 Answer 1

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Let $Q=Q(\pi,it)$ be the analytic conductor of $\pi\otimes|\det|^{it}$. For $\pi$ not self-dual, the method of de la Vallée-Poussin yields the zero-free region $\sigma>1-c/\log Q$ for some effective constant $c>0$ depending on $m$. For $\pi$ self-dual, this is still valid up to a possible exceptional zero $\sigma_0$ that might exist on the real axis within this region. Still, we can bound $\sigma_0<1-cQ^{-C}$ with some effective constants $c,C>0$ depending on $m$.

For more details and some additional conditions under which the exceptional zero does not exist, see Hoffstein-Ramakrishnan: Siegel zeros and cusp forms. See especially Theorem A and Corollary 3.2.

Another very relevant paper is Brumley: Effective multiplicity one on $\mathrm{GL}_N$ and narrow zero-free regions for Rankin-Selberg L-functions. See especially Theorem 5 and Corollary 6 there.

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