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We prove an explicit integral representation -- involving the pullback of asuitable Siegel Eisenstein series -- for the twisted standard $L$-functionassociated to a holomorphic vector-valued Siegel cusp form of degree $n$ andarbitrary level. In contrast to all previously proved pullback formulas in thissituation, our formula involves only scalar-valued functions despite beingapplicable to $L$-functions of vector-valued Siegel cusp forms. The key newingredient in our method is a novel choice of local vectors at the archimedeanplace which allows us to exactly compute the archimedean local integral.  By specializing our integral representation to the case $n=2$ we are able toprove a reciprocity law -- predicted by Deligne's conjecture -- for thecritical special values of the twisted standard $L$-function for vector-valuedSiegel cusp forms of degree 2 and arbitrary level. This arithmetic applicationgeneralizes previously proved critical-value results for the full level case.By specializing further to the case of Siegel cusp forms obtained via theRamakrishnan--Shahidi lift, we obtain a reciprocity law for the criticalspecial values of the symmetric fourth $L$-function of a classical newform.

On the standard $L$-function for $GSp_{2n} \times GL_1$ and algebraicity of symmetric fourth $L$-values for $GL_2$