Conjugate operator of integral operator

Question

Give a integral operator $$ L(\phi) = \int \omega \phi d\mu $$ where $\omega$ is a given function. I want to find the conjugate operator of $L$ in $L^2(\mathbb R^n)$, i.e. find a $L^*$ such that $$ \int \psi L(\phi) d\mu = \int\phi L^*(\psi)~~~~~\forall \psi\in L^2(\mathbb R^n) $$ Then ,how to find it ?