Hecke–Fricke converse theorem for degree-2 L-functions

Let $\chi_0$ be the trivial character modulo $N$. Consider a Dirichlet series that continues to an entire function of order one and is bounded in vertical strips. Let $\xi_f(s)$ denote the completed function appearing in the functional equation, and suppose the series satisfies that functional equation and the Euler product

$$L(s,f)=\prod_p\left(1-a_p p^{-s}+\chi_0(p)p^{k-1-2s}\right)^{-1}.$$

Hecke–Fricke converse theorem. If the Dirichlet series satisfies the stated functional equation and Euler product, then it equals $L(s,f)$ for some

$$f\in S_k(\Gamma_0(N),\chi_0).$$

This is the $L$-function reformulation of the Hecke–Fricke characterization conjecture: analytic continuation, growth, the functional equation, and the Euler product should force the Dirichlet series to arise from a holomorphic cusp form. The source presents this as an equivalent formulation, with no resolution supplied.