# Properties

 Degree 2 Conductor $2^{3}$ Sign $-0.961 + 0.274i$ Motivic weight 6 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−4.62 − 6.52i)2-s − 32.4·3-s + (−21.2 + 60.3i)4-s − 199. i·5-s + (150. + 212. i)6-s + 19.6i·7-s + (492. − 140. i)8-s + 326.·9-s + (−1.29e3 + 920. i)10-s − 924.·11-s + (690. − 1.96e3i)12-s − 1.55e3i·13-s + (128. − 90.9i)14-s + 6.46e3i·15-s + (−3.19e3 − 2.56e3i)16-s + 5.14e3·17-s + ⋯
 L(s)  = 1 + (−0.577 − 0.816i)2-s − 1.20·3-s + (−0.331 + 0.943i)4-s − 1.59i·5-s + (0.695 + 0.982i)6-s + 0.0573i·7-s + (0.961 − 0.274i)8-s + 0.448·9-s + (−1.29 + 0.920i)10-s − 0.694·11-s + (0.399 − 1.13i)12-s − 0.705i·13-s + (0.0467 − 0.0331i)14-s + 1.91i·15-s + (−0.779 − 0.626i)16-s + 1.04·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$8$$    =    $$2^{3}$$ $$\varepsilon$$ = $-0.961 + 0.274i$ motivic weight = $$6$$ character : $\chi_{8} (3, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 8,\ (\ :3),\ -0.961 + 0.274i)$$ $$L(\frac{7}{2})$$ $$\approx$$ $$0.0620303 - 0.443723i$$ $$L(\frac12)$$ $$\approx$$ $$0.0620303 - 0.443723i$$ $$L(4)$$ 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 + (4.62 + 6.52i)T$$
good3 $$1 + 32.4T + 729T^{2}$$
5 $$1 + 199. iT - 1.56e4T^{2}$$
7 $$1 - 19.6iT - 1.17e5T^{2}$$
11 $$1 + 924.T + 1.77e6T^{2}$$
13 $$1 + 1.55e3iT - 4.82e6T^{2}$$
17 $$1 - 5.14e3T + 2.41e7T^{2}$$
19 $$1 + 1.69e3T + 4.70e7T^{2}$$
23 $$1 + 1.92e4iT - 1.48e8T^{2}$$
29 $$1 - 1.65e4iT - 5.94e8T^{2}$$
31 $$1 + 7.55e3iT - 8.87e8T^{2}$$
37 $$1 - 2.89e4iT - 2.56e9T^{2}$$
41 $$1 + 5.21e4T + 4.75e9T^{2}$$
43 $$1 - 5.89e3T + 6.32e9T^{2}$$
47 $$1 + 6.44e4iT - 1.07e10T^{2}$$
53 $$1 + 1.97e5iT - 2.21e10T^{2}$$
59 $$1 - 1.42e5T + 4.21e10T^{2}$$
61 $$1 - 9.64e4iT - 5.15e10T^{2}$$
67 $$1 + 7.52e4T + 9.04e10T^{2}$$
71 $$1 + 5.56e5iT - 1.28e11T^{2}$$
73 $$1 - 2.85e5T + 1.51e11T^{2}$$
79 $$1 - 3.42e5iT - 2.43e11T^{2}$$
83 $$1 + 9.29e5T + 3.26e11T^{2}$$
89 $$1 - 4.34e5T + 4.96e11T^{2}$$
97 $$1 - 6.43e5T + 8.32e11T^{2}$$
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\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

−20.17239350882098367322195051708, −18.35610090068436920822282142813, −17.04324461773209812419233519878, −16.33304776733959239223497508252, −12.90432594182836474830169553867, −12.04150628738107464748147852586, −10.35341239530777941003521327660, −8.412656806276341346209563536903, −5.08955307076141463251605139114, −0.59900738430399026808425440808, 5.83390581301779880453249431197, 7.28196052882993323425064481501, 10.17149663938200605468979743911, 11.33866024450355041823754287163, 14.12461614241560506729601217816, 15.57600905421142132280584557032, 17.05924221878944068214319727118, 18.16850458107949772546400630618, 19.10045606120582166191202488481, 21.77355723222417296547922522008