Does the wizard player sense an aura on it, or not?
The object has no magic aura.
The target of Locate Object is the caster, not the searched object: the range of the spell is Self, the spell is affecting the caster, not the object.
The text of Detect Magic says (emphases mine):
For the duration, you sense the presence of magic within 30 feet of you. If you sense magic in this way, you can use your action to see a faint aura around any visible creature or object in the area that bears magic, and you learn its school of magic, if any.
When Detect Magic is active, one may see a faint aura around the caster of Locate Object, since the caster is bearing the magical effects provided by the spell.