There is no standard head weight for a putter. It can be anything nowadays. The relationship between the length, the head weight, and the overall weight of the putter varies from manufacturer to manufacturer and by model.
Most mass-produced putter heads are weighted for a standard off-the-shelf 35-inch putter and weigh around 330 to 350 grams. In theory the head weight should change to keep the same relationship of head weight to grip weight when you either shorten or lengthen the putter.
For the technically minded the head weight is used to calculate the swingweight of the putter. Swingweight is the relationship between the weight of the grip end of the club and the head end and is measured on a 14-inch fulcrum (balance) scale. It has an impact on your tempo and feel.
There is a school of thought that you should use a heavy putter to putt on slow greens and a light putter to putt on fast greens. There are some putters on the market with adjustable head weights, but to me this is just an added complication.
Shortening the shaft stiffens it and changes the overall weight of the putter, but I have found that the change in feel is not that great. It is far more important that your putter is exactly the right length for you, otherwise you are never going to putt consistently.
If you are not 100% satisfied with the balance of your altered putter, the club fitter can change it to suit you by either adjusting the weight under the grip or by applying lead tape to the putter head.
In summary, if your putter head is too light, you could find yourself flicking at the ball with too quick a tempo. A putter with more head weight will promote a slower and more consistent tempo. However, too heavy a head can lead to less touch.