A .50 cal machinegun can be used to shoot down an aircraft. If it's say a light, slow-moving thing. But a rifle? That's bordering on the impossible (there are recorded, though very rare, incidents where anti-tank rifles took down combat aircraft in WW2).
Are we talking them shooting it down from the ground, or an air intercept? From the ground, if they had a fixed emplacement in the right place, maybe. But just look at the kamakaze attacks on our ships in WWII. Many made it through a barrage of AA fire up to and including 5" air burst rounds to plunge into the decks of ships.
It would have to be real light and real slow. Or the NYPD would have to be real lucky.
The most modern of thier fleet seems to be the Bell 412. Max speed, 140 knots.
A cessna 172 has a cruise speed of 122 knots. Max is higher, depending on a number of factors.
So you have a helo trying to intercept a cessna, with a whopping 18 knot overtake.
I suck at math, but I more than make up for it in experience. It takes awhile for that overtake to happen.