# Properties

 Label 2-768-24.5-c2-0-34 Degree $2$ Conductor $768$ Sign $0.973 + 0.229i$ Analytic cond. $20.9264$ Root an. cond. $4.57454$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.55 − 1.57i)3-s − 1.31·5-s + 10.2·7-s + (4.01 − 8.05i)9-s + 16.6·11-s + 18.7i·13-s + (−3.35 + 2.07i)15-s + 4.38i·17-s + 11.5i·19-s + (26.1 − 16.1i)21-s + 16.7i·23-s − 23.2·25-s + (−2.46 − 26.8i)27-s + 12.5·29-s − 20.3·31-s + ⋯
 L(s)  = 1 + (0.850 − 0.526i)3-s − 0.263·5-s + 1.46·7-s + (0.446 − 0.894i)9-s + 1.51·11-s + 1.44i·13-s + (−0.223 + 0.138i)15-s + 0.257i·17-s + 0.608i·19-s + (1.24 − 0.769i)21-s + 0.728i·23-s − 0.930·25-s + (−0.0912 − 0.995i)27-s + 0.432·29-s − 0.655·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $0.973 + 0.229i$ Analytic conductor: $$20.9264$$ Root analytic conductor: $$4.57454$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{768} (641, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :1),\ 0.973 + 0.229i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$3.109487263$$ $$L(\frac12)$$ $$\approx$$ $$3.109487263$$ $$L(2)$$ 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 + (-2.55 + 1.57i)T$$
good5 $$1 + 1.31T + 25T^{2}$$
7 $$1 - 10.2T + 49T^{2}$$
11 $$1 - 16.6T + 121T^{2}$$
13 $$1 - 18.7iT - 169T^{2}$$
17 $$1 - 4.38iT - 289T^{2}$$
19 $$1 - 11.5iT - 361T^{2}$$
23 $$1 - 16.7iT - 529T^{2}$$
29 $$1 - 12.5T + 841T^{2}$$
31 $$1 + 20.3T + 961T^{2}$$
37 $$1 - 18.5iT - 1.36e3T^{2}$$
41 $$1 + 78.6iT - 1.68e3T^{2}$$
43 $$1 + 36.4iT - 1.84e3T^{2}$$
47 $$1 - 19.9iT - 2.20e3T^{2}$$
53 $$1 - 81.3T + 2.80e3T^{2}$$
59 $$1 - 29.9T + 3.48e3T^{2}$$
61 $$1 - 72.0iT - 3.72e3T^{2}$$
67 $$1 - 56.3iT - 4.48e3T^{2}$$
71 $$1 + 136. iT - 5.04e3T^{2}$$
73 $$1 - 80.8T + 5.32e3T^{2}$$
79 $$1 + 86.0T + 6.24e3T^{2}$$
83 $$1 + 80.4T + 6.88e3T^{2}$$
89 $$1 + 131. iT - 7.92e3T^{2}$$
97 $$1 + 20.4T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$