  A new- and old-form theory for Bessel periods of Saito-Kurokawa
representations is given. We introduce arithmetic subgroups so that a local
Bessel vector fixed by the subgroup indexed by the conductor of the
representation is unique up to scalars. The global Langlands L-function of a
holomorphic Saito-Kurokawa representation coincides with a canonically settled
Piatetski-Shapiro zeta integral of the newform.
