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?