Let be a Dirichlet characters of modulus . Thus is a periodic function of period . It is possible, that for values of restricted by condition , the function may have d period less than .
Definition (Primitive characthers) We say that a Dirichlet character of modulus is imprimitive if restricted by has a period which is less than , Otherwise we say that is primitive.
For maximal clarity, let us write out exactly what we mean restricted has a period . This means: For any integers with and we have .
Definition (Conductor) Given a Dirichlet character of modulus , the conductor of is defined to be the smallest positive integer such that restricted by has a period .
Lemma If is a Dirichlet character of modulus and if is a period of restricted by , then . Hence in particular ,we have .
Lemma For each Dirichlet character of modulus , there is a unique Dirichlet character of of modulus such that
This Dirichlet character is primitive