theorem DirichletCharacter.inv_eq_self_of_sq_eq_one {N : ℕ} {χ : DirichletCharacter ℂ N} (hquad : χ ^ 2 = 1) : χ⁻¹ = χ

A multiplicative quadratic character is self-inverse.

theorem DirichletCharacter.IsPrimitive.completedLFunction_one_sub_quadratic {N : ℕ} [NeZero N] {χ : DirichletCharacter ℂ N} (hχ : χ.IsPrimitive) (hquad : χ ^ 2 = 1) (s : ℂ) : completedLFunction χ (1 - s) = ↑N ^ (s - 1 / 2) * χ.rootNumber * completedLFunction χ s

The primitive functional equation specialized to a quadratic character.

theorem DirichletCharacter.IsPrimitive.norm_rootNumber_of_quadratic {N : ℕ} [NeZero N] {χ : DirichletCharacter ℂ N} (hχ : χ.IsPrimitive) (_hquad : χ ^ 2 = 1) : ‖χ.rootNumber‖ = 1

The root number in the primitive quadratic lane has norm one.