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