***Question*** : are the continuous characters of the form 

 - $\eta : \mathbb{Z}_p^* \to \mathbb{Z}_p^*$, or
 - $\eta : (1+p\mathbb{Z}_p)^{\times} \to \mathbb{Z}_p^*$ (i.e., on the principal units in $\mathbb{Z}_p^*$)

well understood? Can such characters be classified in either case ?

I'm hoping to find an analytic classification ; i.e. to describe such characters as functions, or more precisely, how the functions $z\mapsto z^s$ for $s\in\mathcal{O}_{\mathbb{C}_p}$  'sit' inside the set of characters $\eta : (1 + p\mathbb{Z}_p)^\times \to \mathbb{Z}_p^*$ (i.e., how 'far' is a generic character from some character of this type ?).