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Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight l. We prove a pullback formula for certain Eisenstein series—thus generalizing a construction of Shimura— and use this to derive an explicit integral representation for the degree eight L-function L(s, F x g). This integral representation involves the pullback of a simple Siegel-type Eisenstein series on the unitary group GU(3, 3). As an application, we prove a reciprocity law—predicted by Deligne's conjecture—for the critical special values L(m, F x g) where m ∈ ℤ with 2 ≤ m ≤ l//2―1.

Pullbacks of Eisenstein series from GU(3,3) and criticalL-values for GSp(4) ×GL(2)