T On the absolute convergence of automorphic Dirichlet series

Let F(s) = Sigma(infinity)(n=1) an ns be a Dirichlet series in the axiomatically defined class A#. The class A# is known to contain the extended Selberg class S#, as well as all the L-functions of automorphic forms on GL(n)/K, where K is a number field. Let d be the degree of F(s). We show that Sigma(n<X) vertical bar a(n)vertical bar = Omega(X 1/2 + 1/2d), and hence, that the abscissa of absolute convergence of sigma(a) of F(s) must satisfy sigma a >= 1/2 + 1/2d.