Frequently goes to and stays on zero bars of service

My brand new iPhone 5s will randomly go to zero bars of service and stay there.  Usually the only way I notice this is when I try to make a call and just hear dead silence for a minute.  After this I'll look at screen and realize it has happened again.  Rebooting the phone clears it up until it happens again.

I figured I just lived near a problematic cell tower, but this has happened at home and 35 miles away at work.

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