# Properties

 Degree 2 Conductor $2^{3}$ Sign $0.158 - 0.987i$ Motivic weight 9 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (18.9 + 12.3i)2-s + 67.6i·3-s + (207. + 467. i)4-s + 506. i·5-s + (−834. + 1.28e3i)6-s + 300.·7-s + (−1.83e3 + 1.14e4i)8-s + 1.51e4·9-s + (−6.25e3 + 9.61e3i)10-s − 5.52e4i·11-s + (−3.16e4 + 1.40e4i)12-s − 1.60e5i·13-s + (5.69e3 + 3.70e3i)14-s − 3.42e4·15-s + (−1.75e5 + 1.94e5i)16-s + 3.05e5·17-s + ⋯
 L(s)  = 1 + (0.838 + 0.545i)2-s + 0.482i·3-s + (0.405 + 0.914i)4-s + 0.362i·5-s + (−0.262 + 0.404i)6-s + 0.0472·7-s + (−0.158 + 0.987i)8-s + 0.767·9-s + (−0.197 + 0.304i)10-s − 1.13i·11-s + (−0.440 + 0.195i)12-s − 1.55i·13-s + (0.0396 + 0.0257i)14-s − 0.174·15-s + (−0.670 + 0.741i)16-s + 0.886·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(10-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $0.158 - 0.987i$ motivic weight = $$9$$ character : $\chi_{8} (5, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 8,\ (\ :9/2),\ 0.158 - 0.987i)$ $L(5)$ $\approx$ $1.75053 + 1.49225i$ $L(\frac12)$ $\approx$ $1.75053 + 1.49225i$ $L(\frac{11}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-18.9 - 12.3i)T$$
good3 $$1 - 67.6iT - 1.96e4T^{2}$$
5 $$1 - 506. iT - 1.95e6T^{2}$$
7 $$1 - 300.T + 4.03e7T^{2}$$
11 $$1 + 5.52e4iT - 2.35e9T^{2}$$
13 $$1 + 1.60e5iT - 1.06e10T^{2}$$
17 $$1 - 3.05e5T + 1.18e11T^{2}$$
19 $$1 - 6.34e5iT - 3.22e11T^{2}$$
23 $$1 + 2.25e6T + 1.80e12T^{2}$$
29 $$1 + 1.62e6iT - 1.45e13T^{2}$$
31 $$1 + 2.76e6T + 2.64e13T^{2}$$
37 $$1 + 6.07e6iT - 1.29e14T^{2}$$
41 $$1 + 6.28e6T + 3.27e14T^{2}$$
43 $$1 + 2.76e7iT - 5.02e14T^{2}$$
47 $$1 - 1.38e7T + 1.11e15T^{2}$$
53 $$1 - 1.04e8iT - 3.29e15T^{2}$$
59 $$1 - 3.86e7iT - 8.66e15T^{2}$$
61 $$1 + 1.10e8iT - 1.16e16T^{2}$$
67 $$1 + 1.57e8iT - 2.72e16T^{2}$$
71 $$1 + 1.55e8T + 4.58e16T^{2}$$
73 $$1 - 2.67e8T + 5.88e16T^{2}$$
79 $$1 + 2.31e8T + 1.19e17T^{2}$$
83 $$1 - 5.10e8iT - 1.86e17T^{2}$$
89 $$1 + 3.91e8T + 3.50e17T^{2}$$
97 $$1 + 4.08e8T + 7.60e17T^{2}$$
show less
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−20.48719607485803655867989622728, −18.40296490361267609579902071647, −16.58946820601144498908836277980, −15.46777968170485745061311987556, −14.10862111965734679987565821519, −12.48127846311781118117282410228, −10.51138259987051860361226554797, −7.899257537411288051599426808184, −5.73111269654623798831699249891, −3.53901561045568655763416667963, 1.70993775660762988295087346178, 4.51827021190430141688253419569, 6.89286082018099205740364667346, 9.782463760608644528696436301100, 11.84367830379921795258069708987, 12.97619325500560247861138662292, 14.46219352638226535918885426283, 16.13065382618078950656735641591, 18.25995941481561914849259291070, 19.56923155250795989963608775582