A Dirichlet character modulo $q$ is a function $\chi: \mathbb{Z} \to \mathbb{C}$ satisfying: (1) $\chi(n+q) = \chi(n)$ (periodicity), (2) $\chi(mn) = \chi(m)\chi(n)$ (complete multiplicativity), (3) $\chi(n) = 0$ iff $\gcd(n,q) > 1$. There are exactly $\phi(q)$ Dirichlet characters mod $q$, forming a group under pointwise multiplication isomorphic to $(\mathbb{Z}/q\mathbb{Z})^\times$. The principal character $\chi_0$ satisfies $\chi_0(n) = 1$ for $\gcd(n,q) = 1$.

A character $\chi$ mod $q$ is primitive if it is not induced from a character mod $d$ for any proper divisor $d | q$. The conductor $f_\chi$ is the smallest modulus from which $\chi$ can be induced. Every character mod $q$ is induced from a unique primitive character mod $f_\chi$: $\chi(n) = \chi^*(n)\chi_0(n)$ where $\chi^*$ is primitive mod $f_\chi$ and $\chi_0$ is principal mod $q$.

The Gauss sum associated to $\chi$ mod $q$ is $\tau(\chi) = \sum_{n=1}^q \chi(n) e^{2\pi i n/q}$. For a primitive character $\chi$: $|\tau(\chi)| = \sqrt{q}$ and $\tau(\chi)\tau(\overline{\chi}) = \chi(-1)q$. Gauss sums play the role of "Fourier transforms" for characters and appear in the functional equation of $L(s, \chi)$.