My recommendation is this: when your leg and foot are relaxed, the pedal should be neither completely up nor completely down, but at the point where the dampers start to lift. This point is easy to feel on a grand piano: there is very little force needed to depress the pedal down to it, a definite increase as the dampers start to lift, and even more when the dampers lift completely above the string.
The region between where the dampers are just starting to lift above the string and where they are barely touching it is where partial pedalling happens. Beginners won't be concerned with this, but when you get into more advanced pedalling techniques, it becomes very important to be able to control the movement of the pedal with the utmost subtlety in this region. Completely up and completely down are easy, but here you are constantly making minute adjustments of pressure, based on what your ears are telling you. It thus makes sense that this happens around the point where the foot is in a "neutral" position: the muscles at the back and the front of the shin have more or less the same tension (very little), and are able to control the tiny changes in force involved in "half" or partial pedalling.
Here's a graphic representation of pedal pressure compared to distance travelled (from Yamaha Europe):
When my foot is simply resting on the pedal, it's somewhere in the "Half Pedalling Range". If I have a long passage without pedal, I park my foot on the floor beside the pedal.