  We prove an explicit integral representation -- involving the pullback of a
suitable Siegel Eisenstein series -- for the twisted standard $L$-function
associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and
arbitrary level. In contrast to all previously proved pullback formulas in this
situation, our formula involves only scalar-valued functions despite being
applicable to $L$-functions of vector-valued Siegel cusp forms. The key new
ingredient in our method is a novel choice of local vectors at the archimedean
place which allows us to exactly compute the archimedean local integral.
  By specializing our integral representation to the case $n=2$ we are able to
prove a reciprocity law -- predicted by Deligne's conjecture -- for the
critical special values of the twisted standard $L$-function for vector-valued
Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application
generalizes previously proved critical-value results for the full level case.
By specializing further to the case of Siegel cusp forms obtained via the
Ramakrishnan--Shahidi lift, we obtain a reciprocity law for the critical
special values of the symmetric fourth $L$-function of a classical newform.
