Tatuzawa's theorem for Rankin-Selberg $L$-functions

Let $\pi$ and ${\pi }^{\mathrm{\prime }}$ be unitary cuspidal automorphic representations of $\mathrm{G}\mathrm{L}(n)$ and $\mathrm{G}\mathrm{L}(n^{\mathrm{\prime }})$ over a number field $F$. We establish a new zero-free region for all $\mathrm{G}\mathrm{L}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi \times {\pi }^{\mathrm{\prime }})$, generalizing Tatuzawa’s refinement of Siegel’s work on Dirichlet $L$-functions. As a corollary, we show that for all $\epsilon >0$, there exists an effectively computable constant $c>0$ depending only on $(n,n^{\mathrm{\prime }},[F:\mathbb{Q}],\epsilon )$ such that $L(s,\pi \times {\pi }^{\mathrm{\prime }})$ has at most one zero (necessarily simple) in the region $$\mathrm{R}\mathrm{e}(s)\ge 1-c\mathrm{/}(C(\pi )C({\pi }^{\mathrm{\prime }})(\mathrm{\mid }\mathrm{I}\mathrm{m}(s)\mathrm{\mid }+1){)}^{\epsilon },$$ where $C(\pi )$ and $C({\pi }^{\mathrm{\prime }})$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,\pi \times \tilde{\pi })$ by an idele class character $\chi$ apart from a possible single exceptional zero (necessarily real and simple) that can occur only when $\pi \otimes {\chi }^2=\pi$. This extends earlier work of Humphries and Thorner.