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

Let π be an irreducible cuspidal automorphic representation of GLₙ( A_ Q) with associated L-function L(s, π). We study the behaviour of the partial Euler product of L(s, π) at the center of the critical strip. Under the assumption of the Generalized Riemann Hypothesis for L(s, π) 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.