On primitive $p$-adic Rankin-Selberg $L$-functions

Abstract

In this note, we revisit Hida's construction of $p$-adic Rankin-Selberg $L$-functions by incorporating Jacquet's approach to automorphic $L$-functions on $\mathrm{GL}(2) \times \mathrm{GL}(2)$. This allows us to give a construction of primitive three variable $p$-adic Rankin-Selberg $L$-functions associated with a pair of two primitive Hida families in full generality and prove the functional equation of this $p$-adic Rankin-Selberg $L$-function.