# Properties

 Degree 2 Conductor $2^{3}$ Sign $0.719 + 0.694i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (5.62 − 5.69i)2-s + 8.49·3-s + (−0.753 − 63.9i)4-s + 59.7i·5-s + (47.7 − 48.3i)6-s + 483. i·7-s + (−368. − 355. i)8-s − 656.·9-s + (339. + 335. i)10-s + 1.41e3·11-s + (−6.39 − 543. i)12-s − 3.45e3i·13-s + (2.75e3 + 2.71e3i)14-s + 507. i·15-s + (−4.09e3 + 96.3i)16-s − 3.05e3·17-s + ⋯
 L(s)  = 1 + (0.702 − 0.711i)2-s + 0.314·3-s + (−0.0117 − 0.999i)4-s + 0.477i·5-s + (0.221 − 0.223i)6-s + 1.40i·7-s + (−0.719 − 0.694i)8-s − 0.901·9-s + (0.339 + 0.335i)10-s + 1.06·11-s + (−0.00370 − 0.314i)12-s − 1.57i·13-s + (1.00 + 0.991i)14-s + 0.150i·15-s + (−0.999 + 0.0235i)16-s − 0.622·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $0.719 + 0.694i$ motivic weight = $$6$$ character : $\chi_{8} (3, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 8,\ (\ :3),\ 0.719 + 0.694i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$1.58950 - 0.642019i$$ $$L(\frac12)$$ $$\approx$$ $$1.58950 - 0.642019i$$ $$L(4)$$ 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 + (-5.62 + 5.69i)T$$
good3 $$1 - 8.49T + 729T^{2}$$
5 $$1 - 59.7iT - 1.56e4T^{2}$$
7 $$1 - 483. iT - 1.17e5T^{2}$$
11 $$1 - 1.41e3T + 1.77e6T^{2}$$
13 $$1 + 3.45e3iT - 4.82e6T^{2}$$
17 $$1 + 3.05e3T + 2.41e7T^{2}$$
19 $$1 - 968.T + 4.70e7T^{2}$$
23 $$1 - 3.31e3iT - 1.48e8T^{2}$$
29 $$1 - 2.63e4iT - 5.94e8T^{2}$$
31 $$1 + 2.71e4iT - 8.87e8T^{2}$$
37 $$1 + 3.60e4iT - 2.56e9T^{2}$$
41 $$1 + 6.86e3T + 4.75e9T^{2}$$
43 $$1 - 9.28e4T + 6.32e9T^{2}$$
47 $$1 - 1.59e5iT - 1.07e10T^{2}$$
53 $$1 - 8.66e4iT - 2.21e10T^{2}$$
59 $$1 - 1.28e5T + 4.21e10T^{2}$$
61 $$1 + 1.89e5iT - 5.15e10T^{2}$$
67 $$1 + 3.19e5T + 9.04e10T^{2}$$
71 $$1 + 1.96e5iT - 1.28e11T^{2}$$
73 $$1 + 6.39e4T + 1.51e11T^{2}$$
79 $$1 - 1.64e5iT - 2.43e11T^{2}$$
83 $$1 + 8.02e5T + 3.26e11T^{2}$$
89 $$1 + 5.41e4T + 4.96e11T^{2}$$
97 $$1 + 1.10e6T + 8.32e11T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}