LAUSR

Abstract Let F ( s ) = ∑ n a n / λ n s be a general Dirichlet series which is absolutely convergent on ℜ ( s ) > 1 . Assume that F ( s ) has an analytic continuation and satisfies a growth condition, which gives rise to certain invariants namely the degree d F and conductor α F . In this paper, we show that there are at most 2 d F general Dirichlet series with a given degree d F , conductor α F and residue ρ F at s = 1 . As a corollary, we get that elements in the extended Selberg class with positive Dirichlet coefficients are determined by their degree, conductor and the residue at s = 1 .