manuscripta math. 87, 489 - 499 (1995) manuscripta mathematica ~) Springer-Verlag 1995 Characteristic Twists of a Dirichlet Series for Siegel Cusp Forms W. Kohnen, A. Krieg and J. Sengupta Received February 13, 1995 1 Introduction In [8], Kohnen and Skoruppa introduced a new type of Dirichlet series, which is asso- ciated with the Fourier-Jacobi expansion of a pair f,g of Siegel cusp forms of weight k and degree n = 2. The meromorphic continuation and functional equation under s ~-+ 2k - 2 - s with root number +1 were derived. These results were generalized to Hermitian modular forms in [11] and to Siegel modular forms of arbitrary degree n > 2 by Yamazaki [14] as well as partly in [10]. Recently in the case n = 2 analogous results were obtained for twists with a Dirichlet character X, whenever )C 2 is primitive mod a prime p (cf. [7]). The purpose of the present note is to generalize these results to arbitrary degree, when the index of the Jacobi forms is a number (cf. [4, 5, 13]). We derive the meromorphic continuation for twists with an arbitrary Dirichlet character X and the functional equation with the root number (a• 4, whenever X2 is primitive and G x denotes the Gauss sum at- tached to X. The proof proceeds along the lines of [7] and uses properties of generalized Epstein zeta functions (cf. [2]) As a corollary of our main theorem we include results on twists of the spinor zeta function attached to f by Andrianov, whenever f = g is a simultaneous Hecke eigenform of degree 2, whose first Fourier-Jacobi coefficient is non-zero. 2 Preliminaries Let 7-/. denote the Siegel half-space of degree n, Fn the Siegel modular group of degree n and [F., k]0 the space of Siegel cusp forms of degree n and weight k. Each f 6 [Fn, k]0 possesses a Fourier expansion of the form (i) f(Z) : E af(T)e2"it~(Tz}' T>o where T runs through all positive definite half-integral matrices. Now let n > 1 and consider a decomposition w' , T = , [l m