Dirichlet series
| L(χ,s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
| L(s,χ) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s − 15-s + 16-s − 17-s − 18-s − 19-s + 20-s + 21-s + 22-s + 23-s + 24-s + 25-s − 26-s − 27-s − 28-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr
=\mathstrut & \, \Lambda(\chi,1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr
=\mathstrut & \, \Lambda(1-s,\chi)
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(2011\) |
| \( \varepsilon \) | = | $1$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{2011} (2010, \cdot )$ |
| Sato-Tate | : | $\mu(2)$ |
| primitive | : | yes |
| self-dual | : | yes |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 2011,\ (1:\ ),\ 1)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.5869725302$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.5869725302$ |
| $L(\chi,1)$ | $\approx$ | 0.4903903070 |
| $L(1,\chi)$ | $\approx$ | 0.4903903070 |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]