# Properties

 Label 2-768-12.11-c1-0-10 Degree $2$ Conductor $768$ Sign $1$ Analytic cond. $6.13251$ Root an. cond. $2.47639$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.73i·3-s + 4i·7-s − 2.99·9-s + 6.92·13-s + 3.46i·19-s + 6.92·21-s + 5·25-s + 5.19i·27-s + 4i·31-s + 6.92·37-s − 11.9i·39-s − 10.3i·43-s − 9·49-s + 5.99·57-s + 6.92·61-s + ⋯
 L(s)  = 1 − 0.999i·3-s + 1.51i·7-s − 0.999·9-s + 1.92·13-s + 0.794i·19-s + 1.51·21-s + 25-s + 0.999i·27-s + 0.718i·31-s + 1.13·37-s − 1.92i·39-s − 1.58i·43-s − 1.28·49-s + 0.794·57-s + 0.887·61-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $1$ Analytic conductor: $$6.13251$$ Root analytic conductor: $$2.47639$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{768} (767, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.57067$$ $$L(\frac12)$$ $$\approx$$ $$1.57067$$ $$L(\frac{3}{2})$$ 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 + 1.73iT$$
good5 $$1 - 5T^{2}$$
7 $$1 - 4iT - 7T^{2}$$
11 $$1 + 11T^{2}$$
13 $$1 - 6.92T + 13T^{2}$$
17 $$1 - 17T^{2}$$
19 $$1 - 3.46iT - 19T^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 - 4iT - 31T^{2}$$
37 $$1 - 6.92T + 37T^{2}$$
41 $$1 - 41T^{2}$$
43 $$1 + 10.3iT - 43T^{2}$$
47 $$1 + 47T^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 6.92T + 61T^{2}$$
67 $$1 + 3.46iT - 67T^{2}$$
71 $$1 + 71T^{2}$$
73 $$1 + 10T + 73T^{2}$$
79 $$1 - 4iT - 79T^{2}$$
83 $$1 + 83T^{2}$$
89 $$1 - 89T^{2}$$
97 $$1 - 14T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$