Dave, the well known problem of excessive oil consumption with the LS6 engine as well as all kinds of ECU glitches, traction control, active handling, etc.. The engine also seemed very anemic to me from day one. Just not what I excepted a Z06 should be power wise and I am sure that engine was not up to its full potential...
Problem was I could not even find a Corvette certified mechanic as they were as rare as unicorns for some strange reason! Chevy dealers that sold Corvettes and yet could not service them! Absurd. I am sure it is different now.
I finally had enough when the side passenger window would not go back up as it started to rain and I had to find a shelter from the pouring rain instead of making a business meeting... In a bout of total frustration and outrage over the crap ownership experience I have unleashed on myself with that purchase, I took that car to the first car dealer on the way (BMW) and drove away in a brand new 2002 M3. That turned out to be a fantastic ride which I kept for some years. That E46 gen M3 with the S54 straight-six engine felt quicker and more planted with a better sorted out suspension tuning than the Z06.
I don't dislike Corvettes overall, I have owned a "few". Just never been lucky with them.