Twisting of Siegel modular forms with characters, and $L$-functions

Linear twistings of Siegel modular forms with Dirichlet characters are considered. It is shown that the twisting operators transform modular forms to modular forms. Commutation of twisting operators and Hecke operators is examined. It is proved that under certain conditions the spinor zeta-function of a twisted modular form can be interpreted as the $L$-function of the initial modular form with twisting character. As an illustration of the twist techniques, analytic properties of $L$-functions of cusp forms of genus $n=1$ are considered.