Towards a $\mathrm{GL}_n$ variant of the Hoheisel phenomenon

Let $\pi$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field, and let $\tilde{\pi}$ be contragredient to $\pi$. We prove effective upper and lower bounds of the correct order in the short interval prime number theorem for the Rankin-Selberg $L$-function $L(s,\pi\times\tilde{\pi})$, extending the work of Hoheisel and Linnik. Along the way, we prove for the first time that $L(s,\pi\times\widetilde{\pi})$ has an unconditional standard zero-free region apart from a possible Landau-Siegel zero.