In the classical picture, there is the (complex) modular form, defined on the (complex) upper half plane, which is related to the (complex) $L$-function via the Mellin transform. As I have recently been aware of the notions of $p$-adic modular forms, $p$-adic upper half plane, and $p$-adic $L$-functions, I was thus wondering if there is a $p$-adic analogue of this classical picture which relates the three objects. If not, what would be a 'good/correct/conjectural/etc.' analogue in the $p$-adic setting? Any comments would be appreciated.