Saito-Kurokawa lift (awaiting review)

The Saito-Kurokawa lifting associates to an elliptic modular eigenform $f\in S_{2k}({\rm SL}(2,\mathbb{Z}))$, where $k$ is odd, a Siegel modular form $F$ of degree $2$ and weight $k+1$ with respect to the full symplectic group. The $L$-function of $f$ and the spin $L$-function of $F$ are related by $$ L(s,F)=L(s,f)\zeta(s+1/2)\zeta(s-1/2), $$ where $\zeta$ is the Riemann zeta function. The space spanned by all Saito-Kurokawa lifts is the same as the space spanned by all Maass lifts.