About 5.15pm on May 4, Police Scotland received a report regarding a male who shot and killed a Hen Harrier, near to the B7040 from Leadhills to Elvanfoot.
The birds are protected under the Wildlife and Countryside Act.
"A number of witnesses have come forward, and significant enquiry is underway," said Inspector Martin Speirs.
"I urge anyone who witnessed the incident, or saw a male near the area on a quad bike at the time to come forward.
"I also ask that anyone who has CCTV in the area review their cameras.
"Wanton killing of wildlife in these circumstances is unacceptable.
"However there seems to be some confusion around the appropriate course of action when members of the public have concerns over the conduct of others in the countryside."
He advised the public to consult the Scottish Outdoor Access Code (http://www.outdooraccess-scotland.com/)
"My advice is simple," he said.
"If you have any concerns regarding the behaviour of people in the countryside, particularly where there are animals or you suspect firearms to be present, then call Police Scotland on 101 without delay.
"Please have clear directions to the location ready, to allow Police Officers to get to the area as quickly as possible.
"We will attend and investigate your concerns."