# Dirichlet L-function associated to Kronecker symbol

The Fourier coefficients of the genus 2 Eisenstein series on the Siegel upper half-space are given by sums over Cohen functions. The Cohen function contains as a multiplicative factor a Dirichlet L-function $$L_{D_0}(2-w)$$ associated to the Kronecker symbol $$\left(\frac{\dot{}}{D_0}\right)$$, where $$D_0$$ is the discriminant of the quadratic field $$\mathbb Q(\sqrt D)$$ corresponding to the non-positive half-integral summation index $$T=\begin{pmatrix} a& b/2\\ b/2 & c\end{pmatrix}\in\frac 12\mathbb Z^{2\times 2},$$ $$D=b^2-4ac\leq0$$ and $$w$$ is the weight of the series. The exact formulas are given in arXiv:1502.00557, p. 60 or arXiv:1310.1745, p. 7. The Mathematica function implementing Dirichlet L-series is DirichletL. However, I am unsure how to relate the Kronecker symbol to the modulus and especially index in the argument of DirichletL, especially given that the index seems to depend on the chosen convention. Any help is appreciated!

• Unfortunately I cannot parse your question: the math is over my head, and there is no Mathematica code to help me along. Unless you find somebody with discipline-specific knowledge, you may be severely limiting your chances of getting a good answer. Could you try to explain what you need from MMA? It would be best if you could show us an example of a possible input and output. Consider also that it is unlikely that many will go read two papers and learn about your field, just to write an answer here. – MarcoB May 30 '20 at 22:02
• Thank you for your helpful comment. In my view the two articles are not necessary to understand the question. Perhaps the question is even too mathematical to ask here. I also found a related question here and try to find an answer from there. – El Rafu May 31 '20 at 11:14