What would be the best synthetic engine oil to put into a bmw z4 and how many quartz does it take?

Yes, it could damaged your engine,drain and replace filter with what kind was on before,always use synthetic oil,I take my Toyota tundra to dealer to get oil change,and they scan the whole system for free and give me a print out of the results,if something is wrong,they will tell me and how much or I can fix my self.That's the best place to get oil change,hidden benefits.They told me once my license plate bulb was burnt out,they replaced for a dollar.

It's highly doubtful that non-synthetic would damage your engine.
Synthetic decreases wear and modestly increases mpg. Other than that it's not much else.

A good quality multi-viscosity oil. The best viscosity range depends on the temperature where you live but for most climates, 10W-30 is best for an e36 328. I used 20W-50 for many years in the warm Sacramento climate, but I had some odd performance on cool mornings that I attributed to the old VANOS not moving smoothly with the heavier oil, so I switched to 10W-30 and the problem seems better.
This car came with conventional oil in it. Some e36 owners have switched to synthetic motor oil with no problems. My experience with trying a synthetic (Mobile 1) in my two '97 328s was not good so I switched back to Castrol GTX. When I tried synthetic oil in my two '97 328s and my 91 Volvo, all three cars developed oil leaks almost immediately. However, other owners report no problems at all with synthetic oil.

It should be yellow, on the left hand side of the engine.

The best oil is the Synthetic type oil that one can get through Walmart or an auto parts store.
However, if the car hasn't been using synthetic type, it's a good idea to creep up on it using a mixture of synthetic and regular oil.

However, I have a couple of Cadillac Devilles that use regular oil and I've never had any problems with it.

I wouldn't use synthetic unless there is a sticker that specifically says to use it under the hood.

Most American cars don't need synthetic.

Some Porsches, BMW's, Mercedes Benz and higher end performance cars recommend synthetic for use all of the time.
Good luck on choosing the best grade of oil for your Lincoln LS.

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Every 3000 miles or 6 months which ever comes first--with synthetic 4000 to 5000 miles

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you can substitute with another coolant, but the manufactors one is recommended for many reasons. Best of all because it has been tested for the heatin BMW engines. Mix it 50/50 with water and I'd reccommend using the bmw coolant to stop overheating and causeing serious damage i.e. gead gasket.

10W30 - Total / Elf synthetic oil is the best option. Change every 6000k miles or 10k kilometers.

The Castrol synthetic 5W20 oil may be too light for your car. None of the following three sources recommend 5W20 oil.

Per Bentley manual for 1999 and later E39 BMW (excluding M5), BMW HP 5W30 synthetic oil (07 51 0 017 866) is recommended.
Per the Mobil Oil web site, Mobil 1 synthetic 0W40 oil is recommended.
Per the Castrol Oil web site:
SM 5W-30 - Thank you for inquiring about what oil is best for your vehicle. Based on the information you provided, your vehicle requires a special engine oil that can only be purchased at your local dealer. Please visit your local dealer to ensure that your vehicle is serviced properly.

Does the noise vary or is it constant? Noticed only at startup or all the time?
Another ticking noise source could be the VANOS unit.
PT Stanley

oil consumption is normal on BMW's especially if you drive the car hard and especially if it is still cold.