Iphone 5 wont charge unless turned on??

Hi

I have bought an Iphone 5 second hand and have had the battery replaced, however the previous owner warned me that if the phone turns itself off, you will not be able to charge it again?? I havent risked letting the battery drain down to 0% so dont really know if the new battery i have had replaced in the phone has cured the problem??

I was wondering if anyone else has had issues regarding the problem outlined above??

Thanks in advance