On the absolute convergence of automorphic Dirichlet series

Abstract

Let [Formula: see text] be a Dirichlet series in the axiomatically defined class [Formula: see text]. The class [Formula: see text] is known to contain the extended Selberg class [Formula: see text], as well as all the [Formula: see text]-functions of automorphic forms on [Formula: see text], where [Formula: see text] is a number field. Let [Formula: see text] be the degree of [Formula: see text]. We show that [Formula: see text], and hence, that the abscissa of absolute convergence of [Formula: see text] of [Formula: see text] must satisfy [Formula: see text].