On the Functional Independence of Zeta-Functions of Certain Cusp Forms

The zeta-function $\zeta(s,F)$, $s=\sigma+it$ of a cusp form $F$ of weight $\kappa$ in the half-plane $\sigma>(\kappa+1)/2$ is defined by the Dirichlet series whose coefficients are the coefficients of the Fourier series of the form $F$. The compositions $V(\zeta(s,F))$ with an operator $V$ on the space of analytic functions are considered, and the functional independence of these compositions for certain classes of operators $V$ is proved.