  Let $\pi$ be an irreducible cuspidal automorphic representation of
$\text{GL}_n(\mathbb A_\mathbb Q)$ with associated $L$-function $L(s, \pi)$. We
study the behaviour of the partial Euler product of $L(s, \pi)$ at the center
of the critical strip. Under the assumption of the Generalized Riemann
Hypothesis for $L(s, \pi)$ and assuming the Ramanujan--Petersson conjecture
when necessary, we establish an asymptotic, off a set of finite logarithmic
measure, for the partial Euler product at the central point that confirms a
conjecture of Kurokawa. As an application, we obtain results towards
Chebyshev's bias in the recently proposed framework of Aoki-Koyama.
