Paramodular Böcherer conjecture

Let $F\in S^k(\Gamma^{\text{para}}[p])^+$, where $S^k(\Gamma^{\text{para}}[p])^+$ is the plus space of paramodular cusp forms, and for a fundamental discriminant $D<0$ define

$$A_F(D):=\sum_{\{T>0:\;\operatorname{disc} T=D\}/\hat{\Gamma}_0(p)}\frac{a(T;F)}{\varepsilon(T)},$$

where $\varepsilon(T):=\#\{U\in\hat{\Gamma}_0(p):T[U]=T\}$. Write $A(D)=A_F(D)$ when $F$ is understood, and let $\chi_D$ be the quadratic character associated with $D$. Paramodular Böcherer's Conjecture. For fundamental discriminants $D<0$,

$$L(F,1/2,\chi_D)=\star\,C_F|D|^{1-k}A(D)^2,$$

where $C_F$ is a positive constant depending only on $F$, and $\star=1$ when $p\nmid D$ while $\star=2$ when $p\mid D$. This relates quadratic twists of the spinor $L$-function of a plus-space paramodular form to its averaged Fourier coefficients; the paper proves the formula for Gritsenko lifts and presents numerical evidence for nonlifts.