Is the principal character mod 1 considered primitive?

Question

I am under the impression that the principal Dirichlet character $\chi_0\bmod{q}$ is primitive if and only if $q=1$. However, I read in Davenport's multiplicative number theory that the principal character is "left unclassified in terms of primitivity". So my question is as follows: is the principal Dirichlet character $\chi_0\bmod{1}$ considered primitive?

Answer 1

I'd say it should be, just as $1$ is a primitive first root of unity. Do you have a compelling reason not to call that character primitive? It sure isn't imprimitive. Would we decide not to call a trivial Dirichlet character a Dirichlet character at all because it doesn't have some properties common to nontrivial Dirichlet characters?

Using this convention means some theorems about primitive Dirichlet characters in a book that doesn't use that convention may need to be stated for the "nontrivial primitive characters" rather than just for the "primitive characters". For example, the $L$-function of a nontrivial primitive character is analytic on ${\rm Re}(s) = 1$.