Perform a word study on the word “scepter.” What does the phrase “scepter of righteousness” mean to us? 542 - 2 - 6

There are several occurrences of the word Sceptre in the KJV. Some occurrences refer to the coming Messiah and others to earthly kings/ kingdoms/ regions. Note Scepter is spelled Sceptre in the KJV.

Sceptre
H7626
שׁבט
shêbeṭ
shay'-bet
From an unused root probably meaning to branch off; a scion, that is, (literally) a stick (for punishing, writing, fighting, ruling, walking, etc.) or (figuratively) a clan: -  X correction, dart, rod, sceptre, staff, tribe.

H7626 refers to the coming Messiah in the following verses:

Genesis 49:10, “The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh come; and unto him shall the gathering of the people be.”

Numbers 24:17, “I shall see him, but not now: I shall behold him, but not nigh: there shall come a Star out of Jacob, and a Sceptre shall rise out of Israel, and shall smite the corners of Moab, and destroy all the children of Sheth.”

Psalm 45:6, “Thy throne, O God, is for ever and ever: the sceptre of thy kingdom is a right sceptre.”

H7626 also refers to the defeat of earthy rulers as in verses Isaiah 14:5, “The Lord hath broken the staff of the wicked, and the sceptre of the rulers.” Also Amos 1:5, Amos 1:8 and Zechariah 10:11.

I’m not quite sure how to classify Ezekiel 19:14, “And fire is gone out of a rod of her branches, which hath devoured her fruit, so that she hath no strong rod to be a sceptre to rule. This is a lamentation, and shall be for a lamentation.”

H8275
שׁרביט
sharbîyṭ
shar-beet'
From H7626; a rod of empire: - sceptre.

Here, the reference is to another earthly king married to Esther, the heroine of the Book of Esther. Esther put her life at risk when she approached her husband King Artaxerxes without being summonsed. However, Esther 8:4, “Then the king held out the golden sceptre toward Esther. So Esther arose, and stood before the king,” Also Esther 4:11 and Esther 5:2

G4464
ῥάβδος
rhabdos
hrab'-dos
From the base of G4474; a stick or wand (as a cudgel, a cane or a baton of royalty): - rod, sceptre, staff.

Here used as an affirmation of the righteous rule of Jesus. Hebrews 1:8, “But unto the Son he saith, Thy throne, O God, is for ever and ever: a sceptre of righteousness is the sceptre of thy kingdom.”

Righteousness - H3072
יהוה צדקנוּ
yehôvâh tsidqênû
yeh-ho-vaw' tsid-kay'-noo
From H3068 and H6664 with pronominal suffix.; Jehovah (is) our right; Jehovah-Tsidkenu, a symbolical epithet of the Messiah and of Jerusalem: - the Lord our righteousness.
To me the phrase “scepter of righteousness” refers to a scepter (authority) that belongs to/ was earned by our righteous ruler, our Lord Jesus Christ.

Excuse the pun, but I can’t let go of the scepter without referring to my past efforts in discussing the importance of Genesis 49. Genesis 49:1, “And Jacob called unto his sons, and said, Gather yourselves together, that I may tell you that which shall befall you in the last days.” In part Jacob said to Judah, Genesis 49:8-9, “Judah, thou art he whom thy brethren shall praise: thy hand shall be in the neck of thine enemies; thy father's children shall bow down before thee. Judah is a lion's whelp: from the prey, my son, thou art gone up: he stooped down, he couched as a lion, and as an old lion; who shall rouse him up?" Jesus was the Lion of Judah and in the line of David.

And Jacob continued with the important Genesis 49:10, “The sceptre shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh come; and unto him shall the gathering of the people be.”

Shiloh H7886
שׁילה
shîylôh
shee-lo'
From H7951; tranquil; Shiloh, an epithet of the Messiah: - Shiloh.

In 7 AD Caesar Augustus appointed Caponius Procurator who in turn removed judicial authority from Judea. The scepter refers to tribal identity and self-determination when applying capital punishment, jus gladii, “The right of the sword.” So the temple leaders beat their breasts, etc. and cried that God broke His word in Genesis 49:10; that the scepter had departed Judah.
If the Sanhedrin had really believed God's Word they would have known that Shiloh had come and was a child living in Israel in 7 AD.