The vanishing order of certain Hecke L-functions of imaginary quadratic fields

Let −D<−4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of exists. Let d be a fundamental discriminant prime to D. Let 2k−1 be an odd natural number prime to the class number of . Let χ be the twist of the (2k−1)th power of a canonical Hecke character of by the Kroneckerʹs symbol . It is proved that the vanishing order of the Hecke L-function L(s,χ) at its central point s=k is determined by its root number when , where the constant implied in the symbol depends only on k and ε, and is effective for L-functions with root number −1.