Welcome- moved this thread to a more appropriate forum. It would help to know more about your '03 350Z before trying to guess at a possible answer:
-First, how many miles are on this Z?
-When it first started hesitating, was it in first gear, or was it from a rolling start?
-What have you already done to diagnose this problem?
-Have you pulled codes from the ECM or TCM (if you have a auto)?
-What kind of fuel are you using?
Mileage sounds low, so that's not an issue by itself. You're using the correct type of fuel, so that's not a factor either. If the acceleration hesitates in fifth gear, that's likely a normal reaction. That's 1:1 gearing and the engine has to increase fuel and air delivery from a lean mixture. Since acceleration continues after a brief hesitation, you may be worried about nothing. So pull codes first and let us know what (if anything) you find.
Rene- your battery likely needs replacement. Don't test for voltage while the engine is running, only when the battery is not being energized by the alternator. As for P1564, that code has more to do with the steering wheel wiring. For example, I usually get it from the on-board diagnostic system on my Z racecar because the steering lock system has been pulled and a QR steering wheel hasn't got any of the CC controls.
Good check info and I hate to say it, but keep an eye on that battery. If it's getting that low a charge while driving, there's little doubt the alternator is going. That, in turn, puts a strain on your battery and causes it to go bad over time. Besides that, your battery is five years old and most 12V batteries begin to go bad with 3-6 years. I'd get the alternator checked and likely replaced.
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