Let $\pi$ be an irreducible unitary cuspidal representation of $GL_m({ \Bbb A}_{ \Bbb Q})$ and $L(s,\,\pi)$ be the global $L-$function attached to $\pi$. If ${ \rm Re}(s)>1$, $L(s,\,\pi)$ has a Dirichlet series expression. When $\pi$ is self-contragradient, all the coefficients of Dirichlet series are real. In this note, we shall give non-trivial lower bounds for the number of positive and negative coefficients respectively, which is an improvement on the recent work of Jianya Liu and Jie Wu.
Categories: math.NT