Euler Products at the Centre and Applications to Chebyshev's Bias

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.