\indent 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.