Let $\chi$ be an idele class character over a number field $F$, and let $\pi,\pi'$ be non-dihedral twist-inequivalent cuspidal automorphic representations of $\mathrm{ GL}_2(\mathbb{ A}_F)$. We prove that if $m,n\geq 0$ are integers, $m+n\geq 1$, $F$ is totally real, $\chi$ corresponds with a ray class character, and $\pi,\pi'$ correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin--Selberg $L$-function $L(s,\mathrm{ Sym}^m(\pi)\times(\mathrm{ Sym}^n(\pi')\otimes\chi))$ 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 $\pi'$.
Categories: math.NT