Non-vanishing modulo p of Hecke L-values over imaginary quadratic fields

Let p and q be two distinct odd primes. Let K be an imaginary quadratic field over which p and q are both split. Let Psi be a Hecke character over K of infinity type (k, j) with 0 <= - j < k. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters kappa over K which factor through the Z(q)(2)-extension of K, the p-adic valuation of the algebraic part of the L-value L(kappa Psi<overline>,k+j) is a constant independent of kappa. In addition, when j = 0 and certain technical hypothesis holds, this constant is zero.