As it stands, the question doesn't make sense. A language contains a set of strings, not a TM or a DFA or whatever.
So, the more appropriate question to ask would be to ask the properties of:
$$A_{TM} = \{<M,w> | \, \text{M is a TM which accepts w}\}$$
Now, we by the diagonalisation argument, we know that the given language is undecidable.
We can also prove it is Turing recognisable by creating a Universal TM which does the following:
1. Run the TM on the string $w$. If it ever enters its accept state, accept.
2. If it ever enters its reject, reject.
Thus, we've proven that $A_{TM}$ is both Turing Recognisable and undecidable.
Now, we also know that $L$ is decidable if and only if both $L$ and $L'$ are Turing recognisable.
Making use of this theorem, we can conclude that $\overline{A_{TM}}$ is not Turing recognisable because if it were, it would make $A_{TM}$ decidable.
I hope this helps.