The radiator probably isn't helping unless it's below freezing. Your biggest problem is going to be lighting. Most herb need direct sunlight.
Maybe your kitchen uses natural lighting but getting direct sunlight indoors is difficult.
This is what the experts have to say:
Herbs need as much light as possible, preferably the natural variety
from the sun. Direct sunlight, at least four or five hours a day, is
the minimum herbs need. Even with that, and with plants positioned
close to a south, east or west facing window, I like to recommend
supplying supplemental artificial light from growlights.
If you don't have direct sunlight you should consider either a hydroponic lamp or simply rotating indoor/outdoor pots every few days.
As for watering:
When watering, it is better to water more throughly and less often. A
good principle to follow is to water until water comes out of the
drainage holes in the bottom of the container.
And don't forget about soil:
Inexperienced gardeners may think that if soil in the garden is good
enough to grow herbs or other plants, then the same soil must be good
enough for potted herbs. That’s wrong because of the water drainage
factor. Good garden soil drains well enough in the garden, but in a
container the walls of the container restrict the movement of water in
the soil column. Even in unglazed terra cotta containers it is
important to use a good potting mix that contains amendments such as
perlite or vermiculite or sharp builder’s sand to allow water to move
Quotes taken from this great resource: Growing Herbs Indoors