  We prove that any $q$-automatic completely multiplicative function
$f:\mathbb{N}\to\mathbb{C}$ essentially coincides with a Dirichlet character.
This answers a question of J. P. Allouche and L. Goldmakher and confirms a
conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative
functions. Further, assuming two standard conjectures in number theory, the
methods allows for removing the assumption of completeness.
