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 , where the space 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 of cuspidal Jacobi newforms to . 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 . 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 , the lifting produces a cusp form of degree 2 whose completed Spin *L*-function is given by

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

Schmidt also determined the local components of the automorphic representation associated to a classical newform . 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.

The Jacobi group is defined as

where *H* denotes the integral Heisenberg group. This group is isomorphic to the maximal parabolic subgroup of

where * denotes an arbitrary entry.

We also define the groups

and

where *R* is a ring.

**Definition.** A holomorphic function is called a *Jacobi form of index t and weight k* if the function

called a *completed Jacobi form*, is a Siegel modular form of degree 2 and weight *k* with respect to the parabolic group . That is

for any and the function has the Fourier expansion

We call a *Jacobi cusp form* if there is a strict inequality in the summation. The space of Jacobi forms of index *t* and weight k will be denoted by and its subspace of cusp forms by . The space of *completed Jacobi forms of index t and weight k* and its subspace of *completed Jacobi cusp forms* will be denoted by and , respectively.

Now we define the Hecke-Jacobi ring and its action of the space of Jacobi forms. This is used to define a Hecke-Jacobi newform. This section is based on the results of Gritsenko.

**Definition.** Let be a subgroup of a semigroup *G*. A pair is called a *Hecke pair* if for any the double coset is the union of a finite number of left and right cosets relative to . The *Hecke ring* of the pair is the -invariant subspace of the -vector space consisting of all formal finite linear combinations , where . A representation of on this space is defined by right multiplication

For any with and , their product is defined by

This is well-defined since the product is independent of the choices of representatives and .

To the pair we have the associated Hecke ring , called the *(global) Hecke-Jacobi ring*. The Hecke ring associated to the pair will be called the *local Hecke-Jacobi ring* and denoted by . The Hecke-Jacobi ring can be written as a restricted tensor product

Note that the global and local Hecke-Jacobi rings are not commutative. The Hecke-Jacobi ring acts on . Let , say . The action of *X* on is given by

The subring is defined as the set of elements of of determinant 1, or, equivalently, the Hecke ring for the pair . This ring has the form

Similarly, the ring is defined as the set of elements of of determinant 1.

Certain distinguished elements in these Hecke rings will be useful. Define

and

where . Note that the local Hecke-Jacobi ring can also be described as the subring of generated by the elements , and .

**Definition.** Let . Then is called a *Hecke-Jacobi eigenform* if is an eigenform for all operators . That is,

for some .

Let be the subring of generated by and . Let . If is a Hecke-Jacobi eigenform, then is an eigenform with respect to all elements of . This follows from the identity

Combining some results of Gritsenko on the local factors of the Spin *L*-function of Gritsenko lifts with Schmidt’s local factor results, we obtain the following theorem.

**Theorem.** Let be square-free. Let be a paramodular newform in the span of Gritsenko lifts with . Let be the automorphic representation of associated to *F*.

Case 1: . The local component is a spherical representation with Satake parameters satisfying the equations

Case 2: . If , then the local factor of the associated automorphic representation is . If , the local component is a functorial lift of either the Steinberg St or the twisted Steinberg representation of . It is if one of the following equations holds:

Otherwise it is .

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