Unused iPhone 5 not restarting
Even when you switch off you iPhone the system clock draws power from the battery to keep the time and date function on your phone. Addionally the clock needs much lower voltage to keep working while the kernel on iphone wont work or even display the "charging" graphic until a minimum threshold of voltage is reached by the battery. 

I would suggest charging a minimum of 30 mins from a wall charger.
Also note, the older iphone including the iPhone 5 do not charge with USB 3.0 ports (0.95A) so avoid doing this with a newer laptop with USB 3.0 port as there wont be an indicator to let you know that its actually not charging.
I hope this helps and answers your question.