By definition, a Dirichlet character is primitive if it is not induced by a Dirichlet character of smaller modulus. In particular, the trivial Dirichlet character modulo $1$ (i.e. the constant $1$ function on $\mathbb{Z}$) is primitive.

The above definition is convenient in the sense that every Dirichlet character (including the principal Dirichlet characters) is induced from a unique primitive Dirichlet character. This fact generalizes to automorphic forms, e.g. every Hecke eigenform on the upper half-plane comes from a unique primitive Hecke eigenform, but only when the level $1$ forms are regarded primitive.

By definition, a Dirichlet character is primitive if it is not induced by a Dirichlet character of smaller modulus. In particular, the trivial Dirichlet character modulo $1$ (i.e. the constant function $\mathbb{Z}\to\{1\}$) is primitive.

The above definition is convenient in the sense that every Dirichlet character (including the principal Dirichlet characters) is induced from a unique primitive Dirichlet character. This fact generalizes to automorphic forms, e.g. every Hecke eigenform on the upper half-plane comes from a unique primitive Hecke eigenform, but only when the level $1$ forms are regarded primitive.