# Help me understand this Turing-machine Problem concerning $A_{TM}$

I'm a Physics/C.S. student and have been struggling with this particular problem for a few days now. So the task is as following:

Consider the following languages:

$$\hspace{20pt} L_1 = \{\langle M \rangle \, | \, L(M) = A_{TM}\} \\ \hspace{20pt} L_2 = \{\langle M \rangle \, | \, L(M) = \overline{A_{TM}}\}$$

Are these decidable? Prove your decision.

$$A_{TM}$$ is defined as $$\{\langle M , w\rangle \, | \,\text{M is a TM and M accepts w}\}$$

So first off I'm trying to understand the first language $$L_1$$. What exactly would it mean for $$L_1$$ to be decidable? I understand why $$A_{TM}$$ isn't decidable, but what would a TM that decides $$L_1$$ actually do?

I feel like I already know the solution to the second problem but would like to know whether my gut instinct is right. So since $$\overline{A_{TM}}$$ isn't Turing-recognizable it means that $$M$$ with $$L(M) = \overline{A_{TM}}$$ doesn't exist/is empty. Thus $$L_2 = \emptyset$$ and that's easily decidable.

Solving the language $$L_1$$ means knowing for every TM if it's semi-deciding the $$A_{TM}$$ problem.