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I have a 2002 coupe that has developed a starting issue. After the car has been sitting over night one has to turn the key 5-75 times before the starter engages. Once started and has warmed up it will start (almost) first time. Once it sits for period more than 4-5 hours the problem re-occurs. Only codes available refer to right and left door control modules. Wiring at the starter is secure as well as the main lugs on the starter. Problem occurs with all keys.
I am not necessarily recommending this, but a few years ago I had an identical problem with my 03 Z06, and I chased it in wiring, BCM, etc. for months until I finally gave up and threw a new starter at the problem. The problem hasn't happened since, and I've put 30,000 miles or so on the car since then. The starter looked fine, and worked fine when it worked, but there must have been some kind of intermittent problem inside it.
The STARTER SOLENOID on ALL the C5s are well known to develop this issue. To prove it out,,,, have someone BANG/RAP/TAP the starter with a mechanically agitating device (rubber mallet) and IF the mechanical actuation promotes the starter to work,,,,,, the SOLENOID is most likely defective.
You can replace the solenoid for approx $50. BUT,, you have to remove the starter to replace the solenoid. Most people just install a reman starter for approx 130 bucks!!
Check the starter solenoid for clean and tight electrical connections (FREE) before you replace anything.