Strong Multiplicity One for the Selberg Class

In [7] A. Selberg axiomatized properties expected of L-functions and introduced the “Selberg class” which is expected to coincide with the class of all arithmetically interesting L-functions. We recall that an element F of the Selberg class S satisfies the following axioms. • In the half-plane σ > 1 the function F (s) is given by a Dirichlet series ∑∞n=1 aF (n)n with aF (1) = 1 and aF (n) ≪ǫ n for every ǫ > 0. • There is a natural number mF such that (s− 1)F F (s) extends to an analytic function in the entire complex plane. • There is a function ΦF (s) = QFΓF (s)F (s) where QF > 0 and