  We prove the expected algebraicity property for the critical values of
character twists of the standard $L$-function associated to vector-valued
holomorphic Siegel cusp forms of archimedean type $(k_1, k_2, \ldots, k_n)$,
where $k_n \geq 2n+2$ and all $k_i$ are of the same parity. For the proof, we
use an explicit integral representation to reduce to arithmetic properties of
differential operators on vector-valued nearly holomorphic Siegel cusp forms.
We establish these properties via a representation-theoretic approach.
