Let $\mathbb{F}$ be a local field and let $\pi$ be a principal series representation of $GL_2(\mathbb{F})$ that is $\pi=Ind_B^{GL_2}(\chi_1\otimes\chi_2)$ for two characters $\chi_1$ and $\chi_2$ of the maximal torus. Then I came to know from the book **"Automorphic Forms on Adele group"** by **Gelbert** that due to **Casselman** , it follows that conductor of $\pi=$ (conductor of $\chi_1$)$\times$(xonductor of $\chi_2$).

I want to know if there is similar results for $GL_n(\mathbb{F})$ for $n>2$. That is if $\pi=Ind_B^{GL_n}(\chi_1\otimes\chi_2\otimes\dots\otimes\chi_n)$ is a principal series representation of $GL_n(\mathbb{F})$, then is it true that conductor of $\pi= \prod$(conductor of $\chi_i$)?

Please refer some good paper for this.

Thank you in advance.