This is a tricky question to answer, especially as many people may consider certain days their personal "lucky days," even if it is at odds with what might be suggested by their astrological chart. There is also some disagreement as to the dates for the Western astrological calendar due to the fact that planetary bodies are constantly in motion and changing positions relative to one another. Currently, most people who practice astrology believe that those who are born between the 23rd of August and the 22nd of September are born under the sun sign of Virgo.
This web article by amateur astrologer Keith Abbott suggests that a Virgo's luck may be dependent upon when they were born during the month of Virgo. Beginning with the start date of August 23rd, each week corresponds with a quarter of the moon's cycle. This means that whichever week of Virgo you were born during may correspond to increased luck during the first, second, third, or fourth quarter of the moon. He also mentions that Virgos may be especially lucky on days of the month which are divisible by the number three.
This webpage offers an alternate means of finding out your lucky days in life. The author suggests that when the total sum of numbers for a date is the same as the total sum of numbers for your birthdate, you will have a lucky day. The author also offers dates during each month which may be especially lucky for Virgos-- click the link to check out more detailed descriptions.
Another web astrologer states that Wednesday is an especially lucky day for Virgos.
While it can be a challenging task, if you want to get a really in-depth astrological analysis for your personal relationship with the stars, I recommend having someone draw up your star chart. Alternately, you can use this website to generate one, as long as you know the time and place of your birth.
I would like to mention that I am only knowledgeable about the basics of the traditions of astrology, and there is much variation and disagreement of what is accurate in a practice. I hope that through observation of when you feel most lucky in life, you can determine your "lucky day."