# Properties

 Degree 2 Conductor $2^{3} \cdot 3$ Sign $-0.0336 - 0.999i$ Motivic weight 7 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (4.89 − 10.1i)2-s + (−9 + 45.8i)3-s + (−79.9 − 99.9i)4-s − 97.9·5-s + (423. + 316. i)6-s + 1.54e3i·7-s + (−1.41e3 + 326. i)8-s + (−2.02e3 − 826. i)9-s + (−479. + 999. i)10-s + 1.78e3i·11-s + (5.30e3 − 2.77e3i)12-s + 1.01e4i·13-s + (1.57e4 + 7.58e3i)14-s + (881. − 4.49e3i)15-s + (−3.58e3 + 1.59e4i)16-s − 2.75e4i·17-s + ⋯
 L(s)  = 1 + (0.433 − 0.901i)2-s + (−0.192 + 0.981i)3-s + (−0.624 − 0.780i)4-s − 0.350·5-s + (0.801 + 0.598i)6-s + 1.70i·7-s + (−0.974 + 0.225i)8-s + (−0.925 − 0.377i)9-s + (−0.151 + 0.315i)10-s + 0.404i·11-s + (0.886 − 0.463i)12-s + 1.28i·13-s + (1.53 + 0.738i)14-s + (0.0674 − 0.343i)15-s + (−0.218 + 0.975i)16-s − 1.36i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0336 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0336 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$24$$    =    $$2^{3} \cdot 3$$ $$\varepsilon$$ = $-0.0336 - 0.999i$ motivic weight = $$7$$ character : $\chi_{24} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 24,\ (\ :7/2),\ -0.0336 - 0.999i)$ $L(4)$ $\approx$ $0.736212 + 0.761405i$ $L(\frac12)$ $\approx$ $0.736212 + 0.761405i$ $L(\frac{9}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-4.89 + 10.1i)T$$
3 $$1 + (9 - 45.8i)T$$
good5 $$1 + 97.9T + 7.81e4T^{2}$$
7 $$1 - 1.54e3iT - 8.23e5T^{2}$$
11 $$1 - 1.78e3iT - 1.94e7T^{2}$$
13 $$1 - 1.01e4iT - 6.27e7T^{2}$$
17 $$1 + 2.75e4iT - 4.10e8T^{2}$$
19 $$1 - 1.15e4T + 8.93e8T^{2}$$
23 $$1 + 5.55e4T + 3.40e9T^{2}$$
29 $$1 - 5.47e4T + 1.72e10T^{2}$$
31 $$1 + 7.16e4iT - 2.75e10T^{2}$$
37 $$1 - 2.82e5iT - 9.49e10T^{2}$$
41 $$1 - 4.03e5iT - 1.94e11T^{2}$$
43 $$1 - 4.95e5T + 2.71e11T^{2}$$
47 $$1 - 1.13e6T + 5.06e11T^{2}$$
53 $$1 - 5.72e5T + 1.17e12T^{2}$$
59 $$1 - 1.44e6iT - 2.48e12T^{2}$$
61 $$1 - 1.78e6iT - 3.14e12T^{2}$$
67 $$1 - 1.40e6T + 6.06e12T^{2}$$
71 $$1 + 3.56e6T + 9.09e12T^{2}$$
73 $$1 + 2.22e6T + 1.10e13T^{2}$$
79 $$1 - 5.59e6iT - 1.92e13T^{2}$$
83 $$1 + 3.01e6iT - 2.71e13T^{2}$$
89 $$1 + 5.84e6iT - 4.42e13T^{2}$$
97 $$1 - 6.86e6T + 8.07e13T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}