Dirichlet Character - Primitive Characters and Conductor

Primitive Characters and Conductor

Residues mod N give rise to residues mod M, for any factor M of N, by discarding some information. The effect on Dirichlet characters goes in the opposite direction: if χ is a character mod M, it induces a character χ* mod N for any multiple N of M. A character is primitive if it is not induced by any character of smaller modulus.

If χ is a character mod n and d divides n, then we say that the modulus d is an induced modulus for χ if χ(a)=1 for a coprime to n and 1 mod d: equivalently, χ(a) = χ(b) whenever a, b are congruent mod d and each coprime to n. A character is primitive if there is no smaller induced modulus.

We can formalise differently this by defining characters χ₁ mod N₁ and χ₂ mod N₂ to be co-trained if for some modulus N such that N₁ and N₂ both divide N we have χ₁(n) = χ₂(n) for all n coprime to N: that is, there is some character χ* induced by each of χ₁ and χ₂. This is an equivalence relation on characters. A character with the smallest modulus in an equivalence class is primitive and this smallest modulus is the conductor of the characters in the class.

Imprimitivity of characters can lead to missing Euler factors in their L-functions.