Functions determined by characters are linearly independent? Let $X$ be a set with an action on $\mathbb{Z}/N\mathbb{Z}$. For a Dirichlet character $\chi \pmod N$ we set
$$R(\chi)=\left\{ f:X \to \mathbb{C} ~\mid~ f(l s)=\chi(l)f(s) \text{ for all } l\in \mathbb{N} \text{ with } \gcd(l,N)=1,s\in X \right\}.$$
Are these functions $f\in R(\chi)$ linearly independent?
Edit: If they are not linearly independent, what is the conditions need to be added to these functions to be linearly independent? My target here is to study these functions and find a basis.
Best regards.

Suppose $f$ is in the set, and $g=2f$. Then $g(ls)=2f(ls)=2\chi(l)f(s)=\chi(l)g(s)$, so $g$ is in the set, but $f$ and $g$ are certainly not linearly independent. 
