q-series identities and values of certain L-functions

As usual, define Dedekind's eta-function η(z) by the infinite product In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout) where the series D(q) and E(q) are defined by Here d(n) denotes the number of positive divisors of n. We obtain two infinite families of such identities and describe some consequences for L-functions and partitions. For example, if χ 2 is the Kronecker character for ℚ(), these identities can be used to show that