For any even and primitive Dirichlet character \(\psi\) the authors study properties of the function \[ \eta_\psi(z)=q^{-\tfrac 12 L(-1,\psi)} \prod^\infty_{n=1} (1-q^n)^{\psi(n)}, \] where \(q=\exp(2\pi iz)\) and \(L(s,\psi)\) is the Dirichlet \(L\)-function attached to \(\psi\). When \(\psi= \mathbf{1}\) (the trivial character) \(\eta_\psi(z)\) becomes the classical Dedekind eta function \(\eta(z)\), which has weight \(1/2\). But for \(\psi= \mathbf{1}\) the function \(\eta_\psi(z)\) has weight 0 and is not a product or quotient of eta functions. The authors show that for quadratic characters the functions \(\eta_\psi(z)\) enjoy properties similar to those of \(\eta(z)\).