The books of Luke and Acts are one continous narrative -- starting with Luke and ending with Acts. Since the book of Acts ends with Pauls's first stay in prison in Rome, with no date set for his trial, and the persecutions of Christians by Nero (A.D. 64) not having yet taken place, it could not have been written any later than A.D. 63. Since the book of Acts refers back to the book of Luke, and thus this book was completed prior to the book of Acts, it could not have been completed any later than the early 60's or late 50's.
Encountering the New Testament, Walter A. Elwell / Robert W. Yarbrough, 1998, Baker Books, page 99
A Survey of the New Testament, Third Edition, Robert H. Gundry, 1994, A Survey of the New Testament, Third Edition, Robert H. Gundry, 1994, Zondervan, page 210