Whenever I start my car the oil light stays on for an extra 1 second then goes away never to be seen again. It never turns on while driving. Is this normal? Can someone tell me if there car does this?
my oil level is pretty much right below full mark.

 

What oil filter?

 

The light is designed to alert you to low oil pressure whether caused by low oil volume or pump (<15 psi). Compare the reading on your oil pressure guage (which are notorious for being wonky). Overall if you're starting a cold engine that sounds acceptable.

 

What year is your Z? 03-06 or 07-08? (DE or HR?). If you have an HR, you need to be careful about diagnosing this. It could be a failed Oil Gallery gasket that resides inside the timing case. A symptom of this gasket failure is low oil pressure and could potentially cause catastrophic engine damage if left unchecked. If you have an 03-06 350Z, ignore what I just said and refer back to JHC's comment.
Cheers!
-Icer

 

it's an HR engine. And the light stays on very slightly longer on engine still warm

 

Not sure yet oil was changed by previous owner Castrol not sure filter

 

it's an hr with 46k miles. If I keep the AC off the car idles around 400 rpms and will slightly misfire. Until I put ac back on it goes up to 600 and goes away.
(it also has a solid exhaust leak thats very loud im getting fixed soon, not sure if that's what's causing it.)

 

The light will stay on slightly less than a second on a cold engine. But on warm startups (after car has been used prior maybe 1-3 hrs ago) it will stay on for a full second then leave. I'm running Castrol 5w 30 and live in FL. Thicker oil? Gallery gaskets? Car only has 46k miles HR engine
also if I shut off the AC the car idles around 400 and slightly misfires until I turn it back on and will idle normal around 600 (oil around 14 psi hot engine)

 

Your idle is low. Standard idle (warmed up) with all accessories off should be about 650-750 and a bump to 850ish with AC turned on. At least that has been my own experience. 46K mi is low for a gallery gasket to fail, but this issue might be more age and treatment relevant vs. simple mileage. The oil pressure measurement you stated isn't terribly low when considering your idle RPM is under 500. What kind of reading do you see when cruising on the highway (60-70mph in 6th gear). How about in neutral at 1k rpm (warmed up)?

I suspect your gallery gasket is fine, and this is merely a symptom of the low idle. Consider having your throttle bodies cleaned and also having MAF(s) cleaned. I can't remember if the HR has 2xTBs + 2xMAFs, I think it does, but feel free to set me straight If I'm wrong. You may need to execute the Idle Relearn Procedure after cleaning.
Cheers!
-Icer

 

^^^
Good points, Icer. Twin TBs, twin MAFs.
OP- recommend CRC Electronic Cleaner for MAFs, Berryman TB cleaner. When/If you do this be sure to unplug the device and neg battery cable prior.

Honestly though, I'd do my own oil change with Mobil 1 and proper weights 1st and observe. Gl!