# Properties

 Label 2-768-48.29-c0-0-0 Degree $2$ Conductor $768$ Sign $0.923 - 0.382i$ Analytic cond. $0.383281$ Root an. cond. $0.619097$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 − 0.707i)3-s + 1.41i·7-s + 1.00i·9-s + (1 + i)13-s + (1.00 − 1.00i)21-s − i·25-s + (0.707 − 0.707i)27-s + 1.41·31-s + (−1 + i)37-s − 1.41i·39-s − 1.00·49-s + (−1 − i)61-s − 1.41·63-s + (1.41 + 1.41i)67-s + (−0.707 + 0.707i)75-s + ⋯
 L(s)  = 1 + (−0.707 − 0.707i)3-s + 1.41i·7-s + 1.00i·9-s + (1 + i)13-s + (1.00 − 1.00i)21-s − i·25-s + (0.707 − 0.707i)27-s + 1.41·31-s + (−1 + i)37-s − 1.41i·39-s − 1.00·49-s + (−1 − i)61-s − 1.41·63-s + (1.41 + 1.41i)67-s + (−0.707 + 0.707i)75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$768$$    =    $$2^{8} \cdot 3$$ Sign: $0.923 - 0.382i$ Analytic conductor: $$0.383281$$ Root analytic conductor: $$0.619097$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{768} (65, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 768,\ (\ :0),\ 0.923 - 0.382i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7803578202$$ $$L(\frac12)$$ $$\approx$$ $$0.7803578202$$ $$L(1)$$ 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 + (0.707 + 0.707i)T$$
good5 $$1 + iT^{2}$$
7 $$1 - 1.41iT - T^{2}$$
11 $$1 + iT^{2}$$
13 $$1 + (-1 - i)T + iT^{2}$$
17 $$1 - T^{2}$$
19 $$1 + iT^{2}$$
23 $$1 + T^{2}$$
29 $$1 - iT^{2}$$
31 $$1 - 1.41T + T^{2}$$
37 $$1 + (1 - i)T - iT^{2}$$
41 $$1 + T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 - T^{2}$$
53 $$1 + iT^{2}$$
59 $$1 + iT^{2}$$
61 $$1 + (1 + i)T + iT^{2}$$
67 $$1 + (-1.41 - 1.41i)T + iT^{2}$$
71 $$1 + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 + 1.41T + T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + T^{2}$$
97 $$1 + T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$