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I have a 2001 z06 and the car has zero power. No light no nothing. Once a jump pack is connected it will turnover and start and the voltage hangs around about 13.6 and it is perfectly fine to run and drive, I stop and shut the car off and the power completely disappears again and I need to rejump the car to start it. Sometimes when I do something like open the door the rpms will spike, but the car remains drivable, I have no idea what causes this, any help is greatly appreciated!
Have you confirmed the battery is ok? Sounds like it is bad and has high internal resistance or an internal short. I would get it tested or swap another battery in and see what happens first. As far as the tachometer moving like that when voltage is low from a bad battery all kinds of strange electronic things will happen as things start glitching. I've had mine run down when working inside with the doors open and then when to put the key in and there were relays clicking like a machine gun everywhere because they were too low to engage and would disengage immediately and then repeat... In my other car (04 impala) low battery voltage caused all kinds of codes to be set for the traction system etc at startup because it checks those systems first... Anyway start with the battery first, and don't run it much until you do as it can put strain on the alternator.
Also, my battery got low before where it would not charge up enough running the car on long trips. I had to put it on a charger and let it set for hours before it was back at 100% so that it would start after the car sat for more than a couple hours. I suggest getting a smart charger like a noco genius or something as a good investment. If the car is going to sit for more than 2 weeks put it on a float/smart charger. If the battery is not fully charged the car will still start and drive but will die sitting quicker than 2 weeks I've found. C5's have a lot of draw on the battery when not running. Also make sure to check it for leaks as the computer and wiring harness is below it and acid can cause damage there.
Could be simple as dirty post connections. Start there then battery then grounds, when you jump it it bypasses the battery posts because you are connected directly to the cables.