Let $A$ be an abelian variety over $\mathbb{C}$ and $A^*$ the dual Abelian variety. The class of $A$ and the class of $A^*$ in $\mathcal{M}_{\mathbb{C}}=K_0(Var_\mathbb{C})[\mathcal{L}^{- 1}]$ are same?

In general, is there a pair of varieties (just finite type and separated, integral over $\mathbb{C}$) $(X,Y)$, such that these bounded derived categories are equivalence, but these motivic integrations are not same?

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