A vanishing result for Igusa's p-adic zeta functions with character

Let K be a p-adic field and let f be a K-analytic function on an open and compact subset of K 3 . Let R be the valuation ring of K and let be an arbitrary character of R ◊ . Let Zf, (s) be Igusa’s p-adic zeta function. In this paper, we prove a vanishing result for candidate poles of Zf, (s). This result implies that Zf, (s) has no pole with real part less than 1 if f has no point of multiplicity 2.