I will give a brief review of Artin formalism and its $p$-adic variant. Artin formalism gives a factorization of $L$-functions whenever the associated Galois representation decomposes. I will explain why establishing the $p$-adic Artin formalism (or its algebraic counterpart via the Iwasawa Main Conjectures) is a non-trivial problem when there are no critical $L$-values. In particular, I will focus on the case where the Galois representation arises from a self-Rankin-Selberg product of a newform, and present the results in this direction including the one I obtained in my PhD thesis.