# Properties

 Degree $2$ Conductor $8$ Sign $0.634 + 0.772i$ Motivic weight $20$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (979. + 297. i)2-s − 1.18e4·3-s + (8.72e5 + 5.82e5i)4-s − 4.86e6i·5-s + (−1.16e7 − 3.51e6i)6-s − 3.96e8i·7-s + (6.81e8 + 8.29e8i)8-s − 3.34e9·9-s + (1.44e9 − 4.76e9i)10-s + 2.78e10·11-s + (−1.03e10 − 6.89e9i)12-s − 1.92e11i·13-s + (1.17e11 − 3.88e11i)14-s + 5.75e10i·15-s + (4.21e11 + 1.01e12i)16-s + 1.00e12·17-s + ⋯
 L(s)  = 1 + (0.956 + 0.290i)2-s − 0.200·3-s + (0.831 + 0.555i)4-s − 0.497i·5-s + (−0.191 − 0.0581i)6-s − 1.40i·7-s + (0.634 + 0.772i)8-s − 0.959·9-s + (0.144 − 0.476i)10-s + 1.07·11-s + (−0.166 − 0.111i)12-s − 1.39i·13-s + (0.407 − 1.34i)14-s + 0.0997i·15-s + (0.383 + 0.923i)16-s + 0.499·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(21-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8$$    =    $$2^{3}$$ Sign: $0.634 + 0.772i$ Motivic weight: $$20$$ Character: $\chi_{8} (3, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 8,\ (\ :10),\ 0.634 + 0.772i)$$

## Particular Values

 $$L(\frac{21}{2})$$ $$\approx$$ $$2.78843 - 1.31819i$$ $$L(\frac12)$$ $$\approx$$ $$2.78843 - 1.31819i$$ $$L(11)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-979. - 297. i)T$$
good3 $$1 + 1.18e4T + 3.48e9T^{2}$$
5 $$1 + 4.86e6iT - 9.53e13T^{2}$$
7 $$1 + 3.96e8iT - 7.97e16T^{2}$$
11 $$1 - 2.78e10T + 6.72e20T^{2}$$
13 $$1 + 1.92e11iT - 1.90e22T^{2}$$
17 $$1 - 1.00e12T + 4.06e24T^{2}$$
19 $$1 - 9.04e11T + 3.75e25T^{2}$$
23 $$1 + 5.92e13iT - 1.71e27T^{2}$$
29 $$1 + 4.75e14iT - 1.76e29T^{2}$$
31 $$1 - 5.26e14iT - 6.71e29T^{2}$$
37 $$1 - 5.41e15iT - 2.31e31T^{2}$$
41 $$1 + 1.00e16T + 1.80e32T^{2}$$
43 $$1 + 2.73e16T + 4.67e32T^{2}$$
47 $$1 - 5.52e16iT - 2.76e33T^{2}$$
53 $$1 - 1.63e16iT - 3.05e34T^{2}$$
59 $$1 + 5.08e17T + 2.61e35T^{2}$$
61 $$1 - 2.76e17iT - 5.08e35T^{2}$$
67 $$1 + 7.08e17T + 3.32e36T^{2}$$
71 $$1 + 2.20e18iT - 1.05e37T^{2}$$
73 $$1 - 8.31e18T + 1.84e37T^{2}$$
79 $$1 + 2.46e18iT - 8.96e37T^{2}$$
83 $$1 - 1.22e19T + 2.40e38T^{2}$$
89 $$1 - 3.83e19T + 9.72e38T^{2}$$
97 $$1 - 2.89e19T + 5.43e39T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$