Eta - Product Η ( 7 Τ ) 7 / Η ( Τ )

Let LΦ7(s) be the Dirichlet series associated to the eta-product η(7τ)/η(τ)∈M3(Γ0(7), ε) (here ε(n) := ( n 7 ) = (−7 n ) is the Dirichlet character defined by the residue symbol). We show that LΦ7(s) decomposes into the difference of two L-functions: LΦ7(s) = 1 8 ( L(s, ε)L(s− 2, 1)− L(s− 1, ξ)), where i) L(s, ε) and L(s, 1) are Dirichlet L-functions for the characters ε and 1 modulo 7, respectively, and ii) L(s, ξ) is the L-function for a Hecke character ξ of the imaginary quadratic field Q( √−7). This expression of LΦ7(s) gives a new proof of the non-negativity of the Fourier coefficients of the product η(7τ)/η(τ), conjectured in [S3] and proven by Ibukiyama [I]. We also prove the uniqueness of the above decomposition of LΦ7(s) in a suitable sense.