The thing is as with Vitamin C, most supermarket products just have low doses. So to get an effect i recommend getting pure and high dosed preparates of both, each by itself.
Some OTC products already contain vitC and zinc in proper amounts, e.g. Sambucol (not affiliated).
[1] https://en.m.wikipedia.org/wiki/Zinc_and_the_common_cold
I would not take zinc in the same numbers as Vitamin C - but exactly therefor eI said I would take them apart, because, at least in the German and Spanish Market, you have combination preparates with 80mg Vitramin C and 5mg of Zinc - to achieve the effects I observed in myself you'd need 3x10 of these for the Vitamin C amount, whioch would be a lot of zinc which I would not recommend and have no experience with and no knowledge about the risks.
The articles seem to not agree in doses and helpfulness - one says it must be >= 75mg to actually do something, another that even 90+ mg doesn't do anything. And I believe to remember that already from 25mg/day you can get strange effects in the belly as side effects...