Is it safe to have both lumbar xray and abdominal ct scan with just a 3-week interval?

Yes: The abdominal x-ray ha very low radiation exposure. The ct scan, depending on the make and model has about the equivalent of 250 chest x-rays. Radiation exposure is cumulative, so it adds up no matter when you have the studies. There are many patients with multiple ct scans and the risk of problems with a single ct scan and x-ray is exceedingly low.

Not dangerous: The info provided on the chart if you go to the link below should be very helpful. It will show you the radiation dose for many different radiology exams. http://www.radiologyinfo.org/en/safety/?pg=sfty_xray in general, the risk from the two exams you mention is very low.

Is very safe: As a one time event this is an extremely safe combination.