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Let A = (a0 , a1 , · · · , an−1 ) be a finite sequence of complex numbers with modulus 1 of length n. Let A(z) := a0 + a1 z + · · · + an−1 z n−1 be the unimodular polynomial associated with the sequence A and z := e2πi/n . In this talk, we give an exact formula of the L4 norm for A over the unit circle, namely, if n is an odd positive integer, then n−1 n−1 1X 4 X 4 a 4 kAk4 = |A(ζ )| − 3 |A(ζ a )|2 < A(ζ a )A2 (ζ a ) n a=0 n a=0 n−1 n−1 2 4 X 8 X a 2 a − 3 < A(ζ ) A1 (ζ ) + 3 |A(ζ a )A1 (ζ a )|2 , n a=0 n a=0 where A1 (z) := n−1 X `=0 `a` z ` and A2 (z) := n−1 X `2 a` z ` . `=0 Using this formula, we are able to prove that if an−` = εa` for all 1 ≤ ` < n for some fixed complex number ε with |ε| = 1 and a0 := ε−1/2 , then we have 4 5 kAk44 ≥ n2 − 2n + . 3 3 P a a The main term of lower bound is the best possible and is attained by A(z) := 1 + p−1 a=1 p z , where p· is the Legendre symbol and p is a prime ≡ 1 (mod 4). As a corollary, we can show that if A(z) is a reciprocal polynomial of even degree n − 1, then kAk44 ≥ 53 n2 + O(n3/2 ). Also, our result shows that the largest asymptotic merit factor for reciprocal Littlewood polynomials of even degree is 3/2.