The Critical Order of Certain Hecke L-functions of Imaginary Quadratic Fields

Abstract. Let −D < −4 denote a fundamental discriminant which is either odd or divisible by 8, so that the canonical Hecke character of Q( √ −D) exists. Let d be a fundamental discriminant prime to D. Let 2k− 1 be an odd natural integer prime to the class number of Q( √ −D). Let χ be the twist of the (2k − 1)th power of a canonical Hecke character of Q( √ −D) by the Kronecker’s symbol n 7→ ( d n ). It is proved that the order of the Hecke L-function L(s, χ) at its central point s = k is determined by its root number when |d| ≤ c(ε)D 1 24 or, when |d| ≤ c(ε)D 1 12 and k ≥ 2, where ε > 0 and c(ε) is a constant depending only on ε.