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

Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$, generalizing Tatuzawa's refinement of Siegel's work on Dirichlet $L$-functions. As a corollary, we show that for all $\varepsilon>0$, there exists an effectively computable constant $c>0$ depending only on $(n,n',[F:\mathbb{Q}],\varepsilon)$ such that $L(s,\pi\times\pi')$ has at most one zero (necessarily simple) in the region $$\mathrm{Re}(s)\geq 1-c/(C(\pi)C(\pi')(|\mathrm{Im}(s)|+1))^{\varepsilon},$$ where $C(\pi)$ and $C(\pi')$ are the analytic conductors. A crucial component of our proof is a new standard zero-free region for any twist of $L(s,\pi\times\widetilde{\pi})$ 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.