Euler-Zagier multiple balancing-like L-functions associated to Dirichlet characters

Abstract

In the article we study the analytic continuation of Euler-Zagier multiple shifted balancing-like zeta functions and Euler-Zagier multiple balancing-like $L$-functions associated to Dirichlet characters. Let $\chi_1, \dots, \chi_k$ be the Dirichlet characters of same modulus $t \in\mathbb{N}_{\geq 2}$ and $\chi_0$ be the principal character. The Euler-Zagier multiple shifted balancing-like zeta functions and Euler-Zagier multiple balancing-like $L$-functions are defined as: \begin{eqnarray*} \zeta_{EZB, k}(s_1, \dots, s_k\mid h_1, \dots, h_k)&=&\sum_{m_1, m_2,\dots,m_k=0}^{\infty}\frac{1}{x_{tm_1+h_1}^{s_1}}\cdots \frac{1}{x_{t(m_1+\dots+m_k)+(h_1+\dots+h_k)}^{s_k}} \end{eqnarray*} and  \begin{eqnarray*} \mathcal{L}_{EZB, k}(s_1, \dots, s_k\mid \chi_1, \dots, \chi_k)&=&\sum_{1\leq m_1<m_2<\dots<m_k}\frac{\chi_1(m_1)}{x_{m_1}^{s_1}}\cdots \frac{\chi_k(m_k)}{x_{m_k}^{s_k}}, \end{eqnarray*} where $m_i, h_i \in \mathbb{N}$ for $1\leq i \leq k.$  We also compute a complete list of exact singularities and residues of these functions at poles. We further examine the Euler-Zagier multiple balancing-like $L$-functions associated with quadratic characters at negative integer arguments.