A theorem on analytic strong multiplicity one

Let K be an algebraic number field, and π=⊗πv an irreducible, automorphic, cuspidal representation of GLm(AK) with analytic conductor C(π). The theorem on analytic strong multiplicity one established in this note states, essentially, that there exists a positive constant c depending on ε>0,m, and K only, such that π can be decided completely by its local components πv with norm N(v)<c⋅C(π)^2m+ε.