Upper triangular operators and p-adic L-functions

For a newform f for Γ0(N) of even weight k supersingular at a prime p ≥ 5, by using infinite dimensional p-adic analysis, we prove that the p-adic L-function Lp(f,α; χ) has finite order of vanishing at any character of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde \chi _s \left( x \right) = x^s$\end{document}. In particular, under the natural embedding of ℤp in the group of ℂ*p-valued continuous characters of ℤ*p, the order of vanishing at any point is finite.