The Constitution trumps state law, but it appears you do not understand either, you have just “never heard”. States cannot insert new qualifications for election to the Senate, they can only adopt procedures to certify that candidates are qualified and stipulated by the Constitution. Home Ownership is not a criteria.
What Landrieu is doing is unconstitutional. She is not an inhabitant of Louisiana and is not eligible to stand for election. However, this provisiion of the Constitution is inconvenient for politicians and is therefore largely ignored. Senator Thad Cochran does not live in Mississippi, Senator Pat Roberts does not live in Kansas. Both won their primary elections.
Inhabitancy: Although England repealed Parliament’s residency law in 1774, no delegates spoke against a residency requirement for members of Congress. The qualification first came under consideration on August 6 when the Committee of Detail reported its draft of the Constitution. Article 5, section 3 stated, “Every member of the Senate shall be . . . at the time of his election, a resident of the state from which he shall be chosen.”
On August 8, Roger Sherman moved to strike the word “resident” from the House version of the clause, and insert in its place “inhabitant,” a term he considered to be “less liable to misconstruction.” Madison seconded the motion, noting that “resident” might exclude people occasionally absent on public or private business. Delegates agreed to the term, “inhabitant,” and voted against adding a time period to the requirement. The following day, they amended the Senate qualification to include the word, “inhabitant,” prior to passing the clause by unanimous consent.