This material was commonly used where induction hardening was called out. How the bar got to it's present condition could be rather convoluted as it can be either water or oil quench. I don't have any property or mass effect data for 1045 but I have data for 1040 and 1050, both oil and water quenched. Doing a little interpolation between the two sets of data and curves the material you have appears to have undergone some heat treatment as it doesn’t fall in the normalized or annealed hardness range.
One other thing as 1045 is not considered a through hardening steel, i.e. the reason for the induction hardening label. This make determining the physical properties a little more difficult.
If I remember correctly the bar (2"

we used for shafting was supplied as CR, Q, oil, T @ 1000°F. After machining our material was induction hardened in the bearing areas.
If you can't backtrack the heat number to an MTR or the material is going into a critical component I would scrap it and start anew with a known material. Connecting to an MTR isn’t as hard as it once was because so many suppliers have the data online. If you know who your supplier was put the burden on his back as he has the knowhow and means to track the material.
The only other alternative, if not for a critical part, is to Normalize and Heat treat to a known condition then derive you require data (estimated) from the final hardness. This again is tricky as the material isn't a through hardening steel.