  Let $\pi=\otimes\pi_{v}$ and $\pi^{\prime}=\otimes\pi_{v}^{\prime}$ be two
irreducible, automorphic, cuspidal representations of $GL_{m}(\mathbb{A}_{K})
>.$ Using the logarithmic zero-free region of Rankin-Selberg $L$-function,
Moreno established the analytic strong multiplicity one theorem if at least one
of them is self-contragredient, i.e. $\pi$ and $\pi^{\prime}$ will be equal if
they have finitely many same local components $\pi_{v},\pi_{v}^{\prime},$ for
which the norm of places are bounded polynomially by the analytic conductor of
these cuspidal representations. Without the assumption of the
self-contragredient for $\pi,\pi^{\prime},$ Brumley generalized this theorem by
a a different method, which can be seen as an invariant of Rankin-Selberg
method. In this paper, influenced by Landau's smooth method of Perron formula,
we improved the degree of Brumley's polynomial bound to be $4m+\epsilon.$
