New iPhone 5 randomly shuts off then finally won't turn back on.

I bought a new iphone 5 a little over a week ago. For the past couple days, it would randomly shut off. Yesterday, it finally went blank. All I could get is the Apple logo. When the phone is "on" the screen is like a dimly lit black screen. It won't register in itunes so I tried to put it in recovery mode. When I tried the recovery mode, I get the apple logo, not the itunes logo. Hard reset doesn't work and Toggling the silent switch and hard reset doesn't work either. I know it's a problem with the firmware but I can't even get in to restore it. Please help.