Refurbished I-Phone 5 - Won't Charge

Sooo, after two years with my first iPhone, the screen died and I had to order a replacement, refurbished phone. It just came in today and when I received it and began loading things onto it, it was at 33% charged.

I let the battery die and after I plugged it in to charge it, it has stayed on the same black screen with the red battery in the middle for the past several hours - it recognizes that it is charging because when I unplug the cable, it shows the lightning cable symbol on the bottom until I plug it in again. I've also tried charging it on a computer and iTunes won't recognize the phone.

I'm not sure if this is a common problem and can be fixed simply at home, or if I should take it back in to Apple to see if it's just a defective replacement phone? Any thoughts?

Thanks!