The set of turing machine that stops with input zero is not recursive If M is a Turing Machine, consider the subset of $\mathbb{N}$ defined as: 
$$ \mathrm{ZTM} = \{ x \mid M_x(0)\downarrow \} $$

i.e. the set of 'indexes' of machines that halt on zero. It is possible to use Rice's theorem to prove that this set is r.e. but not recursive.

But I'd like to have a direct proof, using directly some diagonalization argument if possible.
Hint: if you want to know whether $M$ accepts $w$, build a machine that replaces its input with $w$ and then does whatever $M$ does.