Do you consider $L$-functions of elliptic curves over $\mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $\mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.

The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${\rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.