Special Values of the Standard Zeta Functions for Elliptic Modular Forms

Let M and k be positive integers and φ a Dirichlet character modulo M. For a normalized cuspidal Hecke eigenform f of weight k and Nebentypus φ with respect to Γ0(M), and a Dirichlet character χ modulo N, let L(f, s, χ) be the standard zeta function of f twisted by χ. (For the precise definition of the standard zeta function, see the paragraph immediately preceding Theorem 3.3.) The twisted standard zeta function of an elliptic modular form is sometimes called a twisted symmetric-square L function, an important subject in number theory, and is related to many other areas, especially to Galois representations. For examples, see [Doi et al. 98] and [Dummigan 01]. The special values of the standard zeta function are particularly important. To be more precise, assume that k is even, and set