Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $\operatorname{GL}\_{n}(\mathbb{A}\_{\mathbb{Q}})$ gives rise to a primitive L-function.


Is the converse true? That is, assuming Ramanujan conjecture, if a degree $n$ primitive L-function comes from an automorphic representation of $\operatorname{GL}\_{n}(\mathbb{A}\_{\mathbb{Q}})$, is this representation necessarily irreducible and cuspidal?



Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?
Yes, and this is true without the Ramanujan conjecture (but see also Peter Humphries' comment below). 


If $\pi$ is not irreducible, say $\pi=\pi\_1\oplus\pi\_2$, then $L(s,\pi)=L(s,\pi\_1)L(s,\pi\_2)$. 


If $\pi$ is not cuspidal, then by Langlands' theory of Eisenstein series, there is a nontrivial partition $n=\sum n\_j$ and cuspidal irreducible representations $\pi\_j$ of $\mathrm{GL}\_{n\_j}(\mathbb{A}\_\mathbb{Q})$ such that $\pi$ is parabolically induced from $\prod\pi\_j$ on $\prod\mathrm{GL}\_{n\_j}(\mathbb{A}\_\mathbb{Q})$ as a Levi subgroup of $\mathrm{GL}\_n(\mathbb{A}\_\mathbb{Q})$. Then, $L(s,\pi)=\prod L(s,\pi\_j)$.


