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Here's the body cam footage from the moment Indianapolis Colts running back Robert Turbin was popped for weed in Utah earlier this month ... when cops say they found a "rolled joint type thing."

The footage begins after the 26-year-old was pulled over for running a stop sign around 11:20 PM on July 8th ... and cops inform Turbin they are going to search the car because they smelled weed.

The officer later informs Turbin of "a rolled joint-type thing that was found in the center console."

Turbin was not arrested -- instead, he was simply given a citation and was ordered to appear in court later this month.