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$\text{PGSp}_{6}$
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 
$\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of 
$\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group 
$G_{2}$ if its (partial) Spin L-function has a pole at 
$s=1$.
