The quadratic Artin conductor of a motivic spectrum

Given a motivic spectrum $K$ over a smooth proper scheme which is dualizable over an open subscheme, we define its quadratic Artin conductor under some assumptions, and prove a formula relating the quadratic Euler characteristic of $K$, the rank of $K$ and the quadratic Artin conductor. As a consequence, we obtain a quadratic refinement of the classical Grothendieck-Ogg-Shafarevich formula.