For sufficiently large k and l, all zeros of E_k² + E_{2k}, E_k³ + E_{3k}, and E_k E_l + E_{k+l} in the fundamental domain lie on the arc A = {e^{iθ} : π/2 ≤ θ ≤ 2π/3}.
Getz, A generalization of a theorem of Rankin and Swinnerton- Dyer on zeros of modular forms, Proceedings of the American Mathematica l Society 132 (8) (2004) 2221–2231
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Zeros of certain combinations of Eisenstein series of weight 2k, 3k, and k + l
For sufficiently large k and l, all zeros of E_k² + E_{2k}, E_k³ + E_{3k}, and E_k E_l + E_{k+l} in the fundamental domain lie on the arc A = {e^{iθ} : π/2 ≤ θ ≤ 2π/3}.