# Properties

 Degree $2$ Conductor $8$ Sign $0.151 + 0.988i$ Motivic weight $17$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (18.3 + 361. i)2-s + 1.37e4i·3-s + (−1.30e5 + 1.32e4i)4-s + 9.63e4i·5-s + (−4.98e6 + 2.52e5i)6-s − 1.47e7·7-s + (−7.18e6 − 4.69e7i)8-s − 6.09e7·9-s + (−3.48e7 + 1.76e6i)10-s + 6.68e8i·11-s + (−1.82e8 − 1.79e9i)12-s − 1.72e8i·13-s + (−2.70e8 − 5.34e9i)14-s − 1.32e9·15-s + (1.68e10 − 3.45e9i)16-s + 1.92e10·17-s + ⋯
 L(s)  = 1 + (0.0506 + 0.998i)2-s + 1.21i·3-s + (−0.994 + 0.101i)4-s + 0.110i·5-s + (−1.21 + 0.0614i)6-s − 0.968·7-s + (−0.151 − 0.988i)8-s − 0.471·9-s + (−0.110 + 0.00558i)10-s + 0.940i·11-s + (−0.122 − 1.20i)12-s − 0.0586i·13-s + (−0.0490 − 0.967i)14-s − 0.133·15-s + (0.979 − 0.201i)16-s + 0.670·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(18-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8$$    =    $$2^{3}$$ Sign: $0.151 + 0.988i$ Motivic weight: $$17$$ Character: $\chi_{8} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 8,\ (\ :17/2),\ 0.151 + 0.988i)$$

## Particular Values

 $$L(9)$$ $$\approx$$ $$0.380443 - 0.326608i$$ $$L(\frac12)$$ $$\approx$$ $$0.380443 - 0.326608i$$ $$L(\frac{19}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-18.3 - 361. i)T$$
good3 $$1 - 1.37e4iT - 1.29e8T^{2}$$
5 $$1 - 9.63e4iT - 7.62e11T^{2}$$
7 $$1 + 1.47e7T + 2.32e14T^{2}$$
11 $$1 - 6.68e8iT - 5.05e17T^{2}$$
13 $$1 + 1.72e8iT - 8.65e18T^{2}$$
17 $$1 - 1.92e10T + 8.27e20T^{2}$$
19 $$1 + 1.11e11iT - 5.48e21T^{2}$$
23 $$1 + 5.92e11T + 1.41e23T^{2}$$
29 $$1 - 2.95e12iT - 7.25e24T^{2}$$
31 $$1 + 7.53e12T + 2.25e25T^{2}$$
37 $$1 + 2.20e13iT - 4.56e26T^{2}$$
41 $$1 + 8.98e13T + 2.61e27T^{2}$$
43 $$1 + 1.45e13iT - 5.87e27T^{2}$$
47 $$1 + 7.43e13T + 2.66e28T^{2}$$
53 $$1 + 2.84e14iT - 2.05e29T^{2}$$
59 $$1 - 1.09e15iT - 1.27e30T^{2}$$
61 $$1 + 4.14e14iT - 2.24e30T^{2}$$
67 $$1 - 2.20e15iT - 1.10e31T^{2}$$
71 $$1 - 6.67e15T + 2.96e31T^{2}$$
73 $$1 + 1.64e15T + 4.74e31T^{2}$$
79 $$1 + 1.46e16T + 1.81e32T^{2}$$
83 $$1 - 3.05e16iT - 4.21e32T^{2}$$
89 $$1 - 4.48e16T + 1.37e33T^{2}$$
97 $$1 + 3.26e16T + 5.95e33T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$