The definition of the universal Turing machine is
$$ U(\langle M \rangle,p)=M(p) $$
where U is a universal Turing taking two inputs $\langle M \rangle$, the binary encoding of a Turing machine, and $p$ a binary program.
What happens in the case where we feed the null Turing machine to the UTM.
$$ U(0,p)=?U(p)? $$
Will this give out the outputs that are specific to the given formulation of the UTM being run?
Also, is there any kind of general properties of $U(p)$ between the universal Turing machines? My intuition tells me $U(p)$ must be the same answer for all UTMs and all $p$, but I would like confirmation.