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

Let χ be an idele class character over a number field F, and let π,π' be non-dihedral twist-inequivalent cuspidal automorphic representations of GL₂(AF). We prove the following results. 1. If m,n≥ 0 are integers, m+n≥ 1, F is totally real, χ corresponds with a ray class character, and π,π' correspond with primitive non-CM holomorphic Hilbert cusp forms, then the Rankin-Selberg L-function L(s,Symᵐ(π)×(Symⁿ(π')⊗χ)) has a standard zero-free region with no exceptional Landau-Siegel zero. When m,n≥ 1 and m+n≥ 4, this is new even for F=Q. As an application, we establish the strongest known unconditional effective rates of convergence in the Sato-Tate distribution for π and the joint Sato-Tate distribution for π and π'. 2. The Rankin-Selberg L-function L(s,Sym²(π)×(Sym² (π')⊗χ)) has a standard zero-free region with no exceptional Landau-Siegel zero. Until now, this was only known when π=π', π is self-dual, and χ is trivial.