  Let $K$ be an algebraic number field, and $\pi=\otimes\pi_{v}$ an
irreducible, automorphic, cuspidal representation of $\GL_{m}(\mathbb{A}_{K})$
with analytic conductor $C(\pi)$. The theorem on analytic strong multiplicity
one established in this note states, essentially, that there exists a positive
constant $c$ depending on $\varepsilon>0, m,$ and $K$ only, such that $\pi$ can
be decided completely by its local components $\pi_{v}$ with norm $N(v)<c\cdot
C(\pi)^{2m+\varepsilon}.$
