  We show a Siegel-Weil formula in the setting of exceptional theta
correspondence. Using this, together with a new Rankin-Selberg integral for the
Spin L-function of $PGSp_6$ discovered by A. Pollack, we prove that a cuspidal
representation of $PGSp_6$ is a (weak) functorial lift from the exceptional
group $G_2$ if its (partial) Spin L-function has a pole at $s=1$.
