standard zero free region of automorphic L-function on GL(N)

Question

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?

Answer 1

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.