In classical logic, a truth value "paradoxical" is not defined, nor the result of building conjunctions of paradoxical and false statements. Saying the liars paradox is paradoxical does not assign a truth value to it, it declares the inability to assign either true or false as truth value.
So whenever you combine a paradoxical and another statement in any way, classical logic does not define the result.
Alternative to binary logic called "many-valued logic calculi" exist (most common is ternary logic using "unknown" as third value), but there probably isn't any many-valued calculus to apply here as a default, so the answer to your question could be: it depends on the calculus applied. Also i am not aware of any such calculus that would handle "paradoxical" as truth value.
If we imagine a logical calculus in which we could formulate the liars paradox and assign a third truth value "paradox", then likely such a calculus could not generally assign "false" to the conjunction of "false" and "paradoxical", as the self-referential effects could cause new paradoxes.
However, for your example, it would seem most reasonable to expect a new useful multi-valued calculus to resolve the conjunction of the false and the paradoxical parts in your example to false, for the reasons explained in the other answers, same as common ternary logic. Hence the other answers tend towards "false".
However anyone can define a "silly" calculus with many truth values, by which the conjunction of "false" and. "paradoxical" is true, false or paradoxical. It just would not be very useful.