  We establish a result of Bombieri-Vinogradov type for the Dirichlet
coefficients at prime ideals of the standard $L$-function associated to a
self-dual cuspidal automorphic representation $\pi$ of $\mathrm{GL}_n$ over a
number field $F$ which is not a quadratic twist of itself. Our result does not
rely on any unproven progress towards the generalized Ramanujan conjecture or
the nonexistence of Landau-Siegel zeros. In particular, when $\pi$ is fixed and
not equal to a quadratic twist of itself, we prove the first unconditional
Siegel-type lower bound for the twisted $L$-values $|L(1,\pi\otimes\chi)|$ in
the $\chi$-aspect, where $\chi$ is a primitive quadratic Hecke character over
$F$. Our result improves the levels of distribution in other works that relied
on these unproven hypotheses. As applications, when $n=2,3,4$, we prove a
$\mathrm{GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial
bound for a certain $\mathrm{GL}_n\times\mathrm{GL}_2$ shifted convolution sum.
