Lotus are ride and handling experts. They spend hours of testing to select the suspension components, tyre manufacturer and compounds. For previous cars that often means a Lotus-specific version of a tyre and that's also the case for Emira, both the Goodyears and the more track-focused Michelins. So yes, for best performance above 7 degrees you want to use one of these Lotus-specific tyres. People will of course try alternatives and potentially say they are better in some way - I tend to trust the judgement of the team that spent 2 years fine tuning the setup, as tyres are a key component of overall ride and handling.
Lotus has launched the Emira, the latest in a long line of stunning mid-engined cars from the famous UK sports car manufacturer. Lotus has selected the Goodyear Eagle F1 SuperSport as the original equipment tyre for the new model.The Emira has been developed to offer the trademark precision and...
news.goodyear.eu
In winter you're more concerned about safety than performance and R&H. So Lotus tend to recommend a winter tyre that isn't a Lotus-specific design, but they will still recommend a particular brand/model of tyre. I haven't seen anything about recommendations for the Emira but for Evora it was Yokohama W-Drive. You can generally fit any winter tyre of the correct size and rating as an alternative.
Useful info here on winter tyres on an Evora if you're less familiar:
I think people thought me mad. Seeing a mid-engined sports car out in thick snow, especially a rear wheel drive one earned me more looks and interest than if I’d been driving along with the car in flames. Usually this would be well deserved, my Esprit with open differential would spin up a wheel...
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