Shen, Xin2013-09-042013-09-042013-06https://hdl.handle.net/11299/156231University of Minnesota Ph.D. June 2013. Major:Mathematics. Advisor:Prof. Dihua Jiang. 1 computer file (PDF); iv, 115 pages.Tensor L-function is one of the important cases in the Langlands conjecture on the analytic properties of L-functions. Using the method of Rankin-Selberg convolution, Ginzburg, Jiang, Rallis and Soudry found an integral representation of the tensor L-functions for symplectic groups with non-generic representations. In this thesis we calculated the local integrals at the unramified places. First we gave a formula for the Whittaker-Shintani functions for symplectic groups, which is a generalization of the Casselman-Shalika formula for the Whittaker function in the generic case. Then we applied our formula and carried out the unramified calculation. We also investigated the local integrals at the non-archimedean, possibly ramified places and obtain some basic properties, such as convergence, rationalities, and non-vanishing of the local integrals for any given complex numbers.en-USFourier-Jacobi modelL-functionNon-genericSymplectic groupUmramifiedWhittaker-Shintani functionThesis or Dissertation