Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions

Let $\chi$ be an idele class character over a number field $F$, and let $\pi ,{\pi }^{\mathrm{\prime }}$ be non-dihedral twist-inequivalent cuspidal automorphic representations of ${\mathrm{G}\mathrm{L}}_2({\mathbb{A}}_F)$. We prove that if $m,n\ge 0$ are integers, $m+n\ge 1$, $F$ is totally real, $\chi$ corresponds with a ray class character, and $\pi ,{\pi }^{\mathrm{\prime }}$ correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin–Selberg $L$-function $L(s,{\mathrm{S}\mathrm{y}\mathrm{m}}^m(\pi )\times ({\mathrm{S}\mathrm{y}\mathrm{m}}^n({\pi }^{\mathrm{\prime }})\otimes \chi ))$ has a standard zero-free region with no exceptional Landau–Siegel zero. This is new even for $F=\mathbb{Q}$. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato–Tate distribution for $\pi$ and the joint Sato–Tate distribution for $\pi$ and ${\pi }^{\mathrm{\prime }}$.