I was reading the book Introduction to Cyclotomic Fields by Lawrence C. Washington and the conductor of a Dirichlet character is defined as:
An example is also given to make the definition clear.
I was wondering how can we determine the conductor of a Dirichlet character defined for a given mod $n$.
Also, if $n$ is prime, will the conductor be $n$ itself?
A Dirichlet character $\bmod N$ (ie. completely multiplicative $N$ periodic function) is of the form $\psi(n) = \chi(n)1_{\gcd(n,N)=1}$ where $\chi$ is a primitive character $\bmod d$ and $d|N,d< N$. Then $d$ is the conductor. It is the least integer $m>0$ such that $\gcd(n,N)=\gcd(n+m,N)=1\implies \psi(n+m)=\psi(n)$. Then $\chi(n) = 0$ if $\gcd(n,d)> 1$ and $\chi(n)= \psi(n+ad)$ if $\gcd(n,d)=\gcd(n+ad,N)=1$
The next step is to use the CRT to find that $\chi = \prod_j \chi_j$ where $d=\prod_j p_j^{e_j}$ and $\chi_j$ is a primitive character $\bmod p_j^{e_j}$.