iMac Graphite G3 date/time problem

Hi all,
I have an iMac G3 graphite that I bought recently for $30 bucks. Love the little thing, always wanted one since I was in elementary school. The thing is this...my parents have this crazy idea that unplugging everything that's not being used will save on their power bill. I'm in college and I didn't bring up the iMac (i have a MacBook lol), so my iMac was the victim of an unplugging. I come back to find that the date's been reset to 1963 and a time not even close to current time.

My first impulse is to think that it is the CMOS battery that is dead...but if it was dead, wouldn't the computer not start up? Or have my basics of computing gone kaput?

Thanks

If the computer is restarted/shut down, it still holds the time. The "magic" comes when I unplug the computer. Granted I haven't actually done time trials to see if its an instantaneous thing or if, after a certain amount of time unplugged, it reverts to that weird time. I should also say it's the same date/time it reverts to every time it does it...