On the absolute convergence of automorphic Dirichlet series

Let F(s) = n=1∞an ns be a Dirichlet series in the axiomatically defined class #. The class # is known to contain the extended Selberg class S#, as well as all the L-functions of automorphic forms on GLn/K, where K is a number field. Let d be the degree of F(s). We show that n<X|an| = ω(X1 2+1 2d), and hence, that the abscissa of absolute convergence of σa of F(s) must satisfy σa ≥ 1/2 + 1/2d.