  We study the local zeta integrals attached to a pair of generic
representations $(\pi,\tau)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic
field. Through a process of unipotent averaging we produce a pair of
corresponding Whittaker functions whose zeta integral is non-zero, and we
express this integral in terms of the Langlands parameters of $\pi$ and $\tau$.
In many cases, these Whittaker functions also serve as a test vector for the
associated Rankin-Selberg (local) $L$-function.
