The website LMFDB Dirichlet Characters defines the conductor of a Dirichlet character as follows:
The conductor of a Dirichlet character $\chi$ modulo $q$ is the least positive integer $q_1$ dividing $q$ for which $\chi(n+q_1)=\chi(n)$ for all $n$ coprime to $q$.
This definition does not seem correct since for example $\chi_{5,1}=\{1, 1, 1, 1, 0\}$ has conductor $1$ and yet $\chi _{5,1}(4+1)=0\neq 1=\chi _{5,1}(4)$.
Question: Is there a simple way to correct the definition above?
Best Answer
The definition in Apostol is that the conductor of $\chi$ modulo $q$ is the smallest induced modulus $d$. We have $d\mid q$. And $d>0$ is an induced modulus for $\chi$ if and only if $$ \chi(a)=\chi(b) $$ whenever $(a,q)=(b,q)=1$ and $a\equiv b \bmod d$. And $d=1$ is an induced modulus for $\chi$ if and only if $\chi=\chi_1$, the principal character. This applies to your example $\chi_{5,1}$.