The p-adic value of Dirichlet L-function at 1

Question

I'm reading the book An introduction to p-adic L-functions recently. The book can be found in https://warwick.ac.uk/fac/sci/maths/people/staff/cwilliams/lecturenotes/lecture_notes_part_i.pdf. In 4.4, the author claimed that the value of the p-adic L-funtion $L_p(\theta,s)$ at $s=1$ is as following: \begin{equation} L_p(\theta,1)=-(1-\theta(p)p^{-1})\frac{1}{G(\theta^{-1})}\sum_{a\in(\mathbb{Z}/N\mathbb{Z})^{\times}}\theta^{-1}(a)log_p(1-\zeta^a) \end{equation} I followed the book and get the formula above. However, I have a question about the formula. The convergence radius of $log_p(1+x)$ is |x|<1 in $\mathbb{C}_p$, but the p-adic absolute value of $\zeta$ is 1. Where am I wrong?