Congruences for the ratios of Rankin--Selberg $L$-functions

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. We computationally investigate this principle for Rankin–Selberg $L$-functions attached to pairs of holomorphic cuspforms, and formulate a precise conjecture in general.