# Properties

 Label 2-768-12.11-c3-0-18 Degree $2$ Conductor $768$ Sign $0.962 - 0.272i$ 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 + (−1.41 − 5i)3-s + (−23 + 14.1i)9-s − 70.7·11-s − 107. i·17-s + 106i·19-s + 125·25-s + (103. + 95i)27-s + (100. + 353. i)33-s + 56.5i·41-s + 290i·43-s + 343·49-s + (−537. + 152i)51-s + (530 − 149. i)57-s − 325.·59-s + 70i·67-s + ⋯
 L(s)  = 1 + (−0.272 − 0.962i)3-s + (−0.851 + 0.523i)9-s − 1.93·11-s − 1.53i·17-s + 1.27i·19-s + 25-s + (0.735 + 0.677i)27-s + (0.527 + 1.86i)33-s + 0.215i·41-s + 1.02i·43-s + 49-s + (−1.47 + 0.417i)51-s + (1.23 − 0.348i)57-s − 0.717·59-s + 0.127i·67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $0.962 - 0.272i$ Analytic conductor: $$45.3134$$ Root analytic conductor: $$6.73152$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{768} (767, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :3/2),\ 0.962 - 0.272i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.015671145$$ $$L(\frac12)$$ $$\approx$$ $$1.015671145$$ $$L(\frac{5}{2})$$ 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 + (1.41 + 5i)T$$
good5 $$1 - 125T^{2}$$
7 $$1 - 343T^{2}$$
11 $$1 + 70.7T + 1.33e3T^{2}$$
13 $$1 + 2.19e3T^{2}$$
17 $$1 + 107. iT - 4.91e3T^{2}$$
19 $$1 - 106iT - 6.85e3T^{2}$$
23 $$1 + 1.21e4T^{2}$$
29 $$1 - 2.43e4T^{2}$$
31 $$1 - 2.97e4T^{2}$$
37 $$1 + 5.06e4T^{2}$$
41 $$1 - 56.5iT - 6.89e4T^{2}$$
43 $$1 - 290iT - 7.95e4T^{2}$$
47 $$1 + 1.03e5T^{2}$$
53 $$1 - 1.48e5T^{2}$$
59 $$1 + 325.T + 2.05e5T^{2}$$
61 $$1 + 2.26e5T^{2}$$
67 $$1 - 70iT - 3.00e5T^{2}$$
71 $$1 + 3.57e5T^{2}$$
73 $$1 - 430T + 3.89e5T^{2}$$
79 $$1 - 4.93e5T^{2}$$
83 $$1 - 681.T + 5.71e5T^{2}$$
89 $$1 - 1.32e3iT - 7.04e5T^{2}$$
97 $$1 - 1.91e3T + 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$