Let γ, δ be real and z be a complex number. If z2γz + δ = 0 has two distinct roots on the line Re z = 1, then it is necessary that  

1
δ ϵ (0, 1)
2
δ ϵ (-1, 0)
3
|δ| = 1
4
δ ϵ [1, ∞)

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