Dirichlet series
| L(χ,s) = 1 | + 5-s − 7-s − 11-s − 13-s − 17-s + 19-s + 23-s + 25-s + 29-s − 31-s − 35-s − 37-s − 41-s + 43-s + 47-s + 49-s + 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯ |
| L(s,χ) = 1 | + 5-s − 7-s − 11-s − 13-s − 17-s + 19-s + 23-s + 25-s + 29-s − 31-s − 35-s − 37-s − 41-s + 43-s + 47-s + 49-s + 53-s − 55-s − 59-s − 61-s − 65-s + 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & \, \Lambda(\chi,1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & \, \Lambda(1-s,\chi)
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(24\) = \(2^{3} \cdot 3\) |
| \( \varepsilon \) | = | $1$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{24} (11, \cdot )$ |
| Sato-Tate | : | $\mu(2)$ |
| primitive | : | yes |
| self-dual | : | yes |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 24,\ (0:\ ),\ 1)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.7094580614$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.7094580614$ |
| $L(\chi,1)$ | $\approx$ | 0.9358813101 |
| $L(1,\chi)$ | $\approx$ | 0.9358813101 |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]