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 Ψ 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 κ over K which factor through the \(\mathbb{Z}_{q}^{2}\) $${\mathbb{Z}}_q^2$$ -extension of K, the p-adic valuation of the algebraic part of the L-value \(L(\overline{\kappa\Psi},k+j)\) $$L(\overline{\kappa \mathrm{\Psi }},k+j)$$ is a constant independent of κ. In addition, when j = 0 and certain technical hypothesis holds, this constant is zero.