# Properties

 Label 2-768-24.11-c3-0-26 Degree $2$ Conductor $768$ Sign $0.430 - 0.902i$ Analytic cond. $45.3134$ Root an. cond. $6.73152$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.73 + 4.89i)3-s − 16.9·5-s − 17.3i·7-s + (−20.9 + 16.9i)9-s − 29.3i·11-s − 26i·13-s + (−29.3 − 83.1i)15-s + 67.8i·17-s − 107.·19-s + (84.8 − 30i)21-s + 176.·23-s + 162.·25-s + (−119. − 73.4i)27-s − 16.9·29-s + 31.1i·31-s + ⋯
 L(s)  = 1 + (0.333 + 0.942i)3-s − 1.51·5-s − 0.935i·7-s + (−0.777 + 0.628i)9-s − 0.805i·11-s − 0.554i·13-s + (−0.505 − 1.43i)15-s + 0.968i·17-s − 1.29·19-s + (0.881 − 0.311i)21-s + 1.59·23-s + 1.30·25-s + (−0.851 − 0.523i)27-s − 0.108·29-s + 0.180i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $0.430 - 0.902i$ Analytic conductor: $$45.3134$$ Root analytic conductor: $$6.73152$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{768} (383, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :3/2),\ 0.430 - 0.902i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.180518583$$ $$L(\frac12)$$ $$\approx$$ $$1.180518583$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-1.73 - 4.89i)T$$
good5 $$1 + 16.9T + 125T^{2}$$
7 $$1 + 17.3iT - 343T^{2}$$
11 $$1 + 29.3iT - 1.33e3T^{2}$$
13 $$1 + 26iT - 2.19e3T^{2}$$
17 $$1 - 67.8iT - 4.91e3T^{2}$$
19 $$1 + 107.T + 6.85e3T^{2}$$
23 $$1 - 176.T + 1.21e4T^{2}$$
29 $$1 + 16.9T + 2.43e4T^{2}$$
31 $$1 - 31.1iT - 2.97e4T^{2}$$
37 $$1 + 206iT - 5.06e4T^{2}$$
41 $$1 - 305. iT - 6.89e4T^{2}$$
43 $$1 - 93.5T + 7.95e4T^{2}$$
47 $$1 - 117.T + 1.03e5T^{2}$$
53 $$1 - 50.9T + 1.48e5T^{2}$$
59 $$1 - 558. iT - 2.05e5T^{2}$$
61 $$1 - 278iT - 2.26e5T^{2}$$
67 $$1 - 890.T + 3.00e5T^{2}$$
71 $$1 - 58.7T + 3.57e5T^{2}$$
73 $$1 - 422T + 3.89e5T^{2}$$
79 $$1 - 668. iT - 4.93e5T^{2}$$
83 $$1 + 29.3iT - 5.71e5T^{2}$$
89 $$1 - 373. iT - 7.04e5T^{2}$$
97 $$1 + 1.07e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$