On a Rankin-Selberg L-Function over Different Fields

Given two unitary automorphic cuspidal representations π and π′ defined on GLm(E) and GLm'(F), respectively, with E and F being Galois extensions of ℚ, we consider two generalized Rankin-Selberg L-functions obtained by forcefully factoring L(s,π)  and  L(s,π'). We prove the absolute convergence of these L-functions for Re(s)>1. The main difficulty in our case is that the two extension fields may be completely unrelated, so we are forced to work either “downstairs” in some intermediate extension between E∩F and ℚ, or “upstairs” in some extension field containing the composite extension EF. We close by investigating some special cases when analytic continuation is possible and show that when the degrees of the extension fields E and F are relatively prime, the two different definitions give the same generating function