In this talk, we will briefly review Artin formalism and its \(p\)-adic variant. Artin formalism provides a factorization of \(L\)-functions whenever the associated Galois representation decomposes. We will explain why the \(p\)-adic Artin formalism becomes a nontrivial problem when there are no critical \(L\)-values. In particular, we will focus on the case where the Galois representation arises from the Rankin–Selberg product of a newform with itself. We will present the results in this direction, including our result in the case where the modular form in question is \(p\)-non-ordinary, and discuss future directions.