I am not entirely sure whether it's the same for a D5300 as for a D3200, but for that one you can only get a long exposure up to 30 seconds within the camera, and must use a time exposure for longer. There is no "T" setting available in the camera, however, only "B" which leaves the shutter open only as long as you press the button, which is no good for steadiness. The solution is to use the infrared remote. When the camera is set to "B" then the infrared remote provides a true "T" function, in which the shutter opens on the first push of the button, and closes on the second. You can therefore simply open the shutter and it will stay until closed.
Just what exposure you need you may have to experiment with. If the shutter is open for long you'll need a small aperture to avoid overexposure, but some will depend on how bright you want the scene to be.
This will all be done in manual mode, since the camera will indeed overexpose in any automatic metered mode. Make sure you turn off Auto ISO, because if you don't it will go to max and overexpose even in manual mode.
For starters, I'd put it in manual mode for something under 30 seconds (no remote needed) and some middling aperture. Try a shot and see whether it gets close. If it's overexposing but enough car lights are present, stop down the aperture. If there are not enough car lights, but the exposure is OK, increase the exposure time and stop down as needed. At some point you'll find you can go no longer without overexposing, unless you use a neutral density filter, but this is likely only to occur if you need a lot of cars or the surrounding scenery is bright.
A lot of what you do will depend on taste, how bright you want the surrounding scene. If you want true night, it will be harder than if you want a kind of moonlit glow.
You may have to experiment with white balance also, so I'd recommend shooting in raw mode so you can change the white balance later at will. You can then start with Auto, and if it's too yellow adjust it in post.
This is a bit late but on the 5300 you can set the camera into manual mode and set it to bulb (will show a bulb) and in essence lock up the mirror on the camera for as long as you need.
I regularly shoot 2-300sec shots on mine for long exposure shots with ND filters and this works fine. Do suggest a remote for long exposures like this as even with my tripod you have to be careful. However at these lengths you can have some forgiveness.
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Just what exposure you need you may have to experiment with. If the shutter is open for long you'll need a small aperture to avoid overexposure, but some will depend on how bright you want the scene to be.
This will all be done in manual mode, since the camera will indeed overexpose in any automatic metered mode. Make sure you turn off Auto ISO, because if you don't it will go to max and overexpose even in manual mode.
For starters, I'd put it in manual mode for something under 30 seconds (no remote needed) and some middling aperture. Try a shot and see whether it gets close. If it's overexposing but enough car lights are present, stop down the aperture. If there are not enough car lights, but the exposure is OK, increase the exposure time and stop down as needed. At some point you'll find you can go no longer without overexposing, unless you use a neutral density filter, but this is likely only to occur if you need a lot of cars or the surrounding scenery is bright.
A lot of what you do will depend on taste, how bright you want the surrounding scene. If you want true night, it will be harder than if you want a kind of moonlit glow.
You may have to experiment with white balance also, so I'd recommend shooting in raw mode so you can change the white balance later at will. You can then start with Auto, and if it's too yellow adjust it in post.
I regularly shoot 2-300sec shots on mine for long exposure shots with ND filters and this works fine. Do suggest a remote for long exposures like this as even with my tripod you have to be careful. However at these lengths you can have some forgiveness.