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There are two well-known L-functions attached to Siegel modular forms of degree two. These are the Spin and the standard L-function. They have been studied by Andrianov, Böcherer, Shimura, and others. Other L-functions associated to degree two modular forms include the adjoint L-function (degree 10), and two others (degree 14 and degree 16).

The Spin L-function is of particular interest in the cases of paramodular forms of level t and weight k. Examples of paramodular forms can be made using the Gritsenko lift, a generalization of the Saito-Kurokawa lifting. The Gritsenko lift is actually a map defined on Jacobi forms, but it may also be considered a lift of classical newforms \mathcal{S}_{2k-2}^{new-}(\Gamma_0(t)), where the space \mathcal{S}_{2k-2}^{new-}(\Gamma_0(t)) consists of newforms such that the sign in the functional equation of their L-function is -1. Skoruppa and Zagier defined an isomorphism from the space J_{k,t}^{cusp, new} of cuspidal Jacobi newforms to \mathcal{S}_{2k-2}^{new-}(\Gamma_0(t)). So one takes a classical newform in the “minus space,” maps it to the space of Jacobi forms using the inverse of the Skoruppa-Zagier map, then lifts it with the Gritsenko lift to the space of paramodular forms S_k(K(t)). We will also call the composition of these two maps the Gritsenko lift.

Euler factors of the Spin L-function for paramodular forms of square-free level t=N were computed by Schmidt using representation theoretic methods. At the time of his paper, the local Langlands correspondence for GSp(4) was unverified, but Gan and Takeda have since proven its existence and so Schmidt’s results hold. He had shown that for f\in \mathcal{S}_{2k-2}^{new-}(\Gamma_0(N)), the lifting produces a cusp form F\in S_k(K(N)) of degree 2 whose completed Spin L-function is given by

L(s,F, {\rm spin})=\dfrac{1}{4\pi}\left(s-\dfrac{1}{2}\right)Z\left(s+\dfrac{1}{2}\right)Z\left(s-\dfrac{1}{2}\right)L(s,f)

where Z is the completed Riemann zeta function. Moreover, this lifting preserves the Atkin-Lehner eigenvalues, i.e., \eta_pF=\varepsilon_pF for every prime p.

Schmidt also determined the local components of the automorphic representation \pi_f associated to a classical newform f\in\mathcal{S}_{2k-2}^{new-}(\Gamma_0(N)). Moreover, he explained that the Gritsenko lift and similar lifts are predicted by Langlands functoriality. Schmidt also computed the Euler factors of the Spin L-function for the space of paramodular forms of square-free level that are lifts using representation theoretic methods since the lift is functorial. Schmidt and Roberts later determined the possible Euler factors of any paramodular representation, but precise information for the non-square-free level case remained unknown.

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