Welcome to fantasy earth: like Earth in all the same ways, with the exception that the sun you know has been replaced with a ring you don't.
What would the climate of Earth be if the sun was replaced with a ring ("magic", for however much it's functioning matters) with the following properties:
In geosynchronous (not geostationary!) orbit, at 35.800 km-ish height, and so taking 24 hours to orbit fantasy earth.
Orbiting at an inclination of 24 degrees.
Infinitely thin (like I said, magic). Imagine a string wrapped around the Earth and you'll get the idea. This infinitely thin part is flavor, more than anything else. It's a single magic loop of infinitely thin yarn outputting a magical amount of light around the planet.
Radiating light at about 0.007 the degree of brightness of the sun, with the same electromagnetic output of the sun, both in respect of what wavelengths are produced, as well as the irradiance that any individual area experiences, within the range of what's currently experienced by Earth.
I'm assuming that from the perspective of the Earth's surface, the ring would move in a figure 8 pattern, with the net result being a ring weaving around it throughout the day
Half of the northern and southern hemispheres would likely experience night, and half day at the same time—what impact that would have on the climate, I don't know.
I'm assuming that some latitudes would experience a steady degree of irradiance, and so result in more static climate zones, but I'm not 100% certain regarding such things.
Addendum: a very rough estimate is fine, no extensive modeling is required, vibes will suffice.