Modern example of a reciprocity law and intuition behind it

I’m very new to the Langlands program and I was going through the Gauss reciprocity law, Hilbert’s 9th problem, Artin’s reciprocity law which allowed him to identify the Artin’s L-functions with the Hecke L-functions, all the way to the Shimura-Tayinama conjecture and the corresponding reciprocity law.

Could someone provide a modern example of a reciprocity law and how it fits in the Langlands picture? These laws all seem to hinge on the identification of certain eigenvalues with cardinalities of solution sets of equations, which is quite suprising. References are also very welcome. Thank you.