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An LSD will help you with traction; however, the Z33 (and most cars without a crash box) doesn't like speed shifting - particularly the early transmission/diff combination so I would recommend that you drop your shift point a few hundred RPM and finesse/rev match the shift to 2nd.
If you are indeed getting actual tire spin in second VERSUS having the car "just" getting twitchy - sometimes hard to distinguish - due to the shock on the drivetrain, this will help a lot.
But most of all, tires are the usual culprit, as everyone has commented.
OP, an LSD will help with your issue. But better tires will help even more
EDIT: Trav why do you always have to be the one to push them over the edge...
T-Rav is actually on staff here. He is very helpful, and knows the Z33 car well. His job is Vetting new posters to keep the board from ricing out. Pun intended.