(Almost) primitivity of Hecke L-functions

Abstract

Let f be a cusp form of the Hecke space ${\frak M}_0(\lambda,k,\epsilon)$ and let L f be the normalized L-function associated to f. Recently it has been proved that L f belongs to an axiomatically defined class of functions $\bar{\cal S}^\sharp$ . We prove that when λ ≤ 2, L f is always almost primitive, i.e., that if L f is written as product of functions in $\bar{\cal S}^\sharp$ , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if $\lambda\notin\{\sqrt{2},\sqrt{3},2\}$ then L f is also primitive, i.e., that if L f = F 1 F 2 then F 1 (or F 2) is constant; for $\lambda\in\{\sqrt{2},\sqrt{3},2\}$ the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L f belongs to the more familiar extended Selberg class ${\cal S}^\sharp$ is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in ${\cal S}^\sharp$ .