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Clintonville is part of Columbus, so same rules.
You have to petition for a permit, which requires permission from all your neighbors, health department inspection of your setup, submit annual plans showing sketches of the housing and detailed schedule of exactly how and when you do chores for poop removal and sanitation. Your run and coop have to have an "impervious floor" (though it is acceptable to pile dirt on top of the floor of the run if you want your birds to be able to have dust baths). You have to submit annual certifications of health for each bird from a vet. There's a maximum of one hour of foraging per day, only if your chickens are physically incapable of getting out of your yard, and only under direct supervision. You're expected to follow them during the foraging hour to remove all poop from the yard. And to answer your question, maximum 5 chickens, and you're right, no roosters.
Most people find some of these rules a little excessive, so they operate under a "don't ask, don't tell" kind of arrangement. So it helps to live in an area where the neighbors think it's cool that you have chickens.
(Can you imagine if you had to follow the same rules to own a cat or a dog?)
I hope one day someone in Columbus will get things going to petition for more reasonable rules. sigh.