  We establish instances of Beilinson-Tate conjectures for automorphic
representations of $\mathrm{PGSp}_6$ whose Spin $L$-function has a pole at
$s=1$. Using the exceptional theta correspondence between the split group of
type $G_2$ and $\mathrm{PGSp}_6$ and assuming the non-vanishing of a certain
archimedean integral, this allows us to confirm a conjecture of Gross and Savin
on rank $7$ motives of type $G_2$.

Number Theory