The country music star was arrested Thursday in Vinita, OK and booked on an obstruction of investigation charge ... according to online jail records.

TMZ obtained Zach's mug shot, which he took at Craig County Jail, and he's mostly without an expression.

It's unclear what led to his arrest.

On the same day he was arrested, Zach's Instagram account posted a photo of farmland from a car window, with the caption ... "On the road again, gonna go see the birds win."

Zach's an avid Philadelphia Eagles fan and the NFL team plays its season-opening game Sunday against the New England Patriots in Massachusetts ... so it's possible Zach was driving to the game when he got busted.