What is the impedance at the centre of an infinitely long dipole
(in
free space)?

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Its not very different from -

Zin = 120 * Ln( Wavelength / d ) ohms.

where d = conductor diameter, both measured in metres.

Thus, at wavelength = 80 metres with 14 gauge copper wire, input
impedance = 1300 ohms approx.

If you don't believe me, just measure it.
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Reg.