  We proof a Garrett-B\"ocherer decomposition of a vector valued Siegel
Eisenstein series $E_{l,0}^2$ of genus 2 transforming with the Weil
representation of $\text{Sp}_2(\mathbb{Z})$ on the group ring
$\mathbb{C}[(L'/L)^2]$. We show that the standard zeta function associated to a
vector valued common eigenform $f$ for the Weil representation can be
meromorphically continued to the whole $s$-plane and that it satisfies a
functional equation. The proof is based on an integral representation of this
zeta function in terms of $f$ and $E_{l,0}^2$.
