Welcome to the forum.
You have a crank no start condition. The buzzing under the bonnet/hood when the ignition is turned on is normal. Its the idle control valve doing its thing.
You have the EML and service engine soon lights on. There will be trouble codes stored in the engine computer (DME). You will need to read these as they are really good clues to what's wrong with your car. The cheapest scanner to set up is to get the OBDFusion app on your smart phone and to buy the OBD2 adapter, Bluetooth for Android and WiFi for Apple.
Set up OBDFusion like this. https://www.e46fanatics.com/threads/...#post-18380161
1999 2.8L Z3 Roadster,
2000 3.0L Z3 Roadster,
There is only one thing more pleasurable than working on a Z3, that's driving it top down on a fine day.
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