Not sure about your numbers there, but the residual being that low is going to make the Z an expensive car to lease. You can't negotiate the residual value though. You're probably better off buying a Z on a 60 month loan if you're going to put 3k down and want a payment in the mid $400s.
On to my next point, it's generally a bad idea to put a down payment on a lease. You'd be better off putting that 3k in your own bank account and using $77 dollars a month of that 3k to contribute to your monthly payment rather than giving it to the dealership for a lower payment. Or look into MSDs and see if your state offers that.
It's also said that if you can't afford the lease payment without putting a big down payment, then you probably shouldn't lease that car.
If you do end up putting 3k down and something happens to the Z, like you end your lease early, or car gets totaled out and insurance pays for it, you'll most likely never see any part of that 3k again. Thus, keep your 3k deposit in your possession and use it to supplement your monthly lease payments if you must.
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2015 Lexus IS 350 AWD F Sport PW/RR
2010 Nissan 370Z Base PW M6 -- sold
2003 Nissan 350Z Touring MB M6 -- sold
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