TOPICS
Search

Rankin L-Series

The Rankin L-series (or Rankin-Selberg L-series) combines the Fourier coefficients of two modular eigenforms. For full-level normalized modular eigenforms f and g with normalized Hecke eigenvalues lambda_f(n) and lambda_g(n), and for Res>1, it is

L(f×g^_,s)=zeta(2s)sum_(n=1)^infty(lambda_f(n)lambda_g(n)^_)/(n^s).

Here, zeta(s) is the Riemann zeta function. For higher levels, the local factors at primes dividing the levels require corresponding modifications. The Rankin-Selberg integral method supplies an Euler product, analytic continuation, and a functional equation. In the diagonal case f=g, the series has a simple pole at s=1.

The Rankin L-series L(f,chi,s) occurring in the Gross-Zagier formula is obtained by pairing a modular newform f with the theta series associated with a group character chi of the class group.

See also

Analytic Continuation, Dirichlet Series, Euler Product, Fourier Coefficient, Functional Equation, Gross-Zagier Formula, Hecke Eigenvalue, Local Factor, Modular Eigenform, Modular Newform, Riemann Zeta Function, Simple Pole, Theta Series

Explore with Wolfram|Alpha

References

Rankin, R. A. "Contributions to the Theory of Ramanujan's Function tau(n) and Similar Arithmetical Functions. II. The Order of the Fourier Coefficients of Integral Modular Forms." Proc. Cambridge Philos. Soc. 35, 357-372, 1939. https://doi.org/10.1017/S0305004100021101.Selberg, A. "Bemerkungen über eine Dirichletsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist." Arch. Math. Naturvid. 43, 47-50, 1940.

Cite this as:

Weisstein, Eric W. "Rankin L-Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RankinL-Series.html

Subject classifications