I am interested in computations with characters on $PGL(2, F)$ and not in $GL(2, F)$, and have some issues concerning definitions.

The notion of conductor is standard for characters $\chi$ of a $p$-adic number field $F=F_p$, and is defined as the least $r$ such that $\chi$ is trivial on $1+p^r\mathcal{O}_p$. Since characters of $GL(2, F)$ are $\chi \circ \det$ for characters $\chi$ on $F$, it is enough to define conductors for characters. 

I am wondering about the analogous definitions for $PGL(2, F)$. Characters on $PGL(2, F)$ can be seen as characters on $GL(2, F)$ trivial on the center. Do we know their form, as $\chi \circ \det$ for specific king of $\chi$ (trivial on squares?). What is the relation between the conductor of $\chi \circ \det$ and the one of $\chi$ in this setting?