On the absolute convergence of automorphic Dirichlet series

Let $F(s)={\sum }_{n=1}^{\mathrm{\infty }}\frac{a_n}{n^s}$ be a Dirichlet series in the axiomatically defined class ${\mathfrak{A}}^{\mathrm{\#}}$ . The class ${\mathfrak A}^{#}$isknowntocontaintheextendedSelbergclass${\mathcal S}^{#}$, as well as all the $L$-functions of automorphic forms on $G{L}_n\mathrm{/}K$, where $K$ is a number field. Let $d$ be the degree of $F(s)$. We show that ${\sum }_{n<X}\mathrm{\mid }{a}_n\mathrm{\mid }=\mathrm{\Omega }(X^{\frac{1}{2}+\frac{1}{2d}})$, and hence, that the abscissa of absolute convergence of ${\sigma }_a$ of $F(s)$ must satisfy ${\sigma }_a\ge 1\mathrm{/}2+1\mathrm{/}2d$.