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.