Galois averages of Rankin-Selberg L-functions

I will first review the notion of Galois averages of Rankin-Selberg L-functions, in particular those of Rankin-Selberg L-functions of weight-two cusp forms times theta series associated to Hecke characters of imaginary quadratic fields. I will then present a conjecture about the behaviour of these averages with the conductor of the character, of which the nonvanishing theorems of Rohrlich, Vatsal and Cornut-Vatsal are special cases. Finally, I will explain a strategy of proof, at least in the setting where the class number is equal to one.