(Almost) primitivity of Hecke L-functions

Let f be a cusp form of the Hecke space M0(λ, k, ) and let Lf be the normalized L-function associated to f . Recently it has been proved that Lf belongs to an axiomatically defined class of functions S̄. We prove that when λ ≤ 2, Lf is always almost primitive, i.e., that if Lf is written as product of functions in S̄, then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if λ 6∈ { √ 2, √ 3, 2} then Lf is also primitive, i.e., that if Lf = F1F2 then F1 (or F2) is constant; for λ ∈ { √ 2, √ 3, 2} the factorization of nonprimitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which Lf belongs to the more familiar extended Selberg class S is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in S.