My iPhone 5s is frozen with iTunes symbol and plug in picture with arrow My phone is stuck with the image of charger and cord, up arrow and iTunes icon - I can't turn iPhone on or off it seems. Will it restart on its own?  I tried holding the power button down for the ten seconds and then just the home key but that doesn't work either.
Your iPhone is stuck in Recovery Mode. Connect your phone to a computer with iTunes and do a hard reset on it (hold down both your home and sleep button for several seconds).
From a small article on getting in and out of Recovery Mode:

To get out of recovery mode, make sure your USB cable is connected to both the laptop running iTunes and your device.You may need to hold the home button while connecting your USB cable as above.
Now hold the home and sleep/wake buttons together until you see the Apple logo appear. Don’t let go until then – it can take several seconds: the recovery graphic will disappear and a black screen might be shown for a while, but keep holding the buttons down until the white Apple appears.