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

Algebraic cycles and functorial lifts from $G_2$ to $\mathrm{PGSp}_6$