Sorry about your trouble, especially so soon after buying your Z.
When the guy at AutoZone did a scan, did you get the actual code that he read? If so, what was it? Also, what was the code that the dealership got when they scanned your Z? I believe the code for a #5 cylinder misfire is P0305. Also, do you have any mods?
I'm sure others will chime in, but just wondering, specifically, what code (or codes) your getting.
In the meantime, here's a thread regarding a #5 cylinder misfiring -
Took to Dealership - Royal Purple.