Let $F(s)=\sum_{ n=1}^{ \infty}\frac{ a_n}{ n^s}$ be a Dirichlet series in the axiomatically defined class ${ \mathfrak A}^{ \#}$ . The class ${ \mathfrak A}^{ \#}$ is known to contain the extended Selberg class ${ \mathcal S}^{ \#}$, 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 $\sum_{ n<X}|a_n|=\Omega(X^{ \frac{ 1}{ 2}+\frac{ 1}{ 2d}})$, and hence, that the abscissa of absolute convergence of $\sigma_a$ of $F(s)$ must satisfy $\sigma_a\ge 1/2+1/2d$.
Categories: math.NT