On the boundedness of Euler-Stieltjes constants for the Rankin-Selberg \(L-\)function

Abstract

Let \(E\) be a Galois extension of \(\mathbb{Q}\) of finite degree and let \(\pi \) and \(\pi'\) be two irreducible automorphic unitary cuspidal representations of \(GL_m(\mathbb{A}_E)\) and \(GL_{m'}(\mathbb{A}_E)\), respectively. Let \(\Lambda(s,\pi\times\widetilde{\pi}')\) be a Rankin-Selberg \(L-\)function attached to the product \(\pi\times\widetilde{\pi}'\), where \(\widetilde{\pi}'\) denotes the contragredient representation of \(\pi'\), and let its finite part (excluding Archimedean factors) be \(L(s,\pi\times\widetilde{\pi}')\). The Euler-Stieltjes constants of the Rankin-Selberg \(L-\)function are the coefficients in the Laurent (Taylor) series expansion around \(s=1+it_0\) of the function \(L(s, \pi \times \widetilde{\pi}')\). In this paper, we derive an upper bound for these constants.