Homepage banner size and load time

Hi, I am having a banner designed for the homepage and want to know what size would be good for it in pixels length and weight for both laptop and desktop. The illustrator needs to know and I am just trying to manage the process before handing over to a developer. Eventually it will be a gif or some files that enables some movement. If you have thoughts on how a coder can work with the file so that it loads faster this would be helpful. Also, I think there is a way for a coder to specify that the image be static sometimes. I might want the banner image to be static on mobile phones as the load time needs to be quick and have the movement be visible on the desktop version. Also, desktop images are more rectangular and the mobile device tend to be a square image, so this needs to be considered. For example, is a desktop is designed say at 1688 px wide or at 1366 px wide which I believe is common what would be a good length- I don’t want it to have such a great length, so that it is hard to see what is below the banner. Also, I prefer a banner that is not too long (length) so that there is still something showing just below the banner. Any thoughts, direction is most appreciated!!

Website banner is the main imagery that takes top billing on your homepage. The size of your banner is going to determine what else you can fit above the fold.