Degree 1 Elements of the Selberg

In [5] A. Selberg axiomatized properties expected of L-functions and introduced the “Selberg class.” We recall that an element F of the Selberg class S satisfies the following axioms. Axiom 1. In the half-plane σ > 1 the function F (s) is given by an absolutely convergent Dirichlet series ∑∞ n=1 a(n)n −s with a(1) = 1 and a(n) ≪ n for every ǫ > 0. Axiom 2. There is a natural number m such that (s − 1)F (s) extends to an analytic function in the entire complex plane. Axiom 3. There is a function Φ(s) = QG(s)F (s) where Q > 0 and