(Almost) primitivity of Hecke $L$-functions

Let $f$ be a cusp form of the Hecke space ${\mathfrak M}_0(\lambda,k,\epsilon)$ 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 $\bar\Selberg^\sharp$. We prove that when $\lambda\leq 2$, $L_f$ is always almost primitive, i.e., that if $L_f$ is written as product of functions in $\bar\Selberg^\sharp$, then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if $\lambda\not\in\{\sqrt{2}, sqrt{3},2\}$ then $L_f$ is also primitive, i.e., that if $L_f = F_1F_2$ 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 $\Selberg^\sharp$ is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in $\Selberg^\sharp$.