On Li’s coefficients for the Rankin–Selberg L-functions

Abstract

We define a generalized Li coefficient for the L-functions attached to the Rankin–Selberg convolution of two cuspidal unitary automorphic representations π and π′ of \(GL_{m}(\mathbb{A}_{F})\) and \(GL_{m^{\prime }}(\mathbb{A}_{F})\) . Using the explicit formula, we obtain an arithmetic representation of the n th Li coefficient \(\lambda _{\pi ,\pi ^{\prime }}(n)\) attached to \(L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime })\) . Then, we deduce a full asymptotic expansion of the archimedean contribution to \(\lambda _{\pi ,\pi ^{\prime }}(n)\) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of \(L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime })\) , the nth Li coefficient \(\lambda _{\pi ,\pi ^{\prime }}(n)\) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μπ(v,j) of π. Namely, we prove that under GRH for \(L(s,\pi _{f}\times \widetilde{\pi}_{f})\) one has \(|\mathop {\mathrm {Re}}\mu_{\pi}(v,j)|\leq \frac{1}{4}\) for all archimedean places v at which π is unramified and all j=1,…,m.