On some p-adic power series attached to the arithmetic of Q(ζp)

Let p be an odd prime number and let θ be a nontrivial even character of the Galois group of Q(ζp)/Q. We prove that the derivative of the Iwasawa power series f(T,θ) is not congruent to zero modulo p, where f(T,θ) is associated to the p-adic L-function Lp(s,θ).