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Let $\pi$ be an irreducible unitary cuspidal representation of $GL_m({\BbbA}_{\Bbb Q})$ 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 $\pi$ isself-contragradient, all the coefficients of Dirichlet series are real. In thisnote, we shall give non-trivial lower bounds for the number of positive andnegative coefficients respectively, which is an improvement on the recent workof Jianya Liu and Jie Wu.

A note on the number of coefficients of automorphic $L-$functions for $GL_m$ with same signs