Do we have a left adjoint of $i^*$ for a closed immersion $i$?

Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.

My questions is: can we construct a left adjoint of $i^*$ in this case? It seems that the left adjoint is not given in the general construction of Grothendieck's six functors.

Edit: Using Serre functors, I can find a left adjoint of $i^*$ in the case that Y is smooth Calabi-Yau and $X$ is also smooth. But I wonder if there is an answer without the Calabi-Yau condition.