Quadratic Euler Characteristic of Geometrically Cyclic Branched Coverings

For an $n$-fold geometrically cyclic branched covering $Y$ of a smooth, projective scheme $X$ branched at a smooth closed subscheme $Z\subset X$ with $n \in k^\times$, we compute the quadratic Euler characteristic of $Y$ in terms of certain Euler classes on $X$ and $Z$ using the quadratic Riemann-Hu