Let us restrict to automorphic representations of $\mathrm{GL}_n$ over $\mathbb{Q}$ with arbitrary $n$ and unitary central character.

If $\pi$ is an irreducible cuspidal representation, then $L(s,\pi)$ is entire unless $\pi\cong|\det|^{it}$ in which case $L(s,\pi)=\zeta(s+it)$ has a simple pole at $s=1-it$. Let us call these $L$-functions cuspidal.

If $\pi=\pi_1\boxplus\dotsb\boxplus\pi_r$ is an isobaric sum of irreducible cuspidal representations, then $L(s,\pi)$ factors uniquely into cuspidal $L$-functions as $L(s,\pi_1)\dotsb L(s,\pi_r)$. See Liu-Ye: Weighted Selberg orthogonality and uniqueness of factorization of automorphic $L$-functions, Forum Math. 17 (2005), 493-512.

If $\pi$ is a general automorphic representation, then $L(s,\pi)$ differs from a product of cuspidal $L$-functions by finitely Euler factors. See Jacquet: Principal $L$-functions of the linear group, In: Automorphic forms, representations and $L$-functions, Part 2, 63-86, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.