Properties

 Degree $2$ Conductor $8$ Sign $0.999 + 0.0259i$ Motivic weight $17$ Primitive yes Self-dual no Analytic rank $0$

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Dirichlet series

 L(s)  = 1 + (−362. − 3.13i)2-s + 1.38e4i·3-s + (1.31e5 + 2.27e3i)4-s − 6.65e5i·5-s + (4.34e4 − 5.02e6i)6-s − 7.57e6·7-s + (−4.74e7 − 1.23e6i)8-s − 6.31e7·9-s + (−2.08e6 + 2.40e8i)10-s − 1.97e8i·11-s + (−3.14e7 + 1.81e9i)12-s − 4.74e9i·13-s + (2.74e9 + 2.37e7i)14-s + 9.22e9·15-s + (1.71e10 + 5.95e8i)16-s + 3.37e10·17-s + ⋯
 L(s)  = 1 + (−0.999 − 0.00866i)2-s + 1.22i·3-s + (0.999 + 0.0173i)4-s − 0.761i·5-s + (0.0105 − 1.22i)6-s − 0.496·7-s + (−0.999 − 0.0259i)8-s − 0.489·9-s + (−0.00659 + 0.761i)10-s − 0.278i·11-s + (−0.0211 + 1.22i)12-s − 1.61i·13-s + (0.496 + 0.00430i)14-s + 0.929·15-s + (0.999 + 0.0346i)16-s + 1.17·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0259i)\, \overline{\Lambda}(18-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0259i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$8$$    =    $$2^{3}$$ Sign: $0.999 + 0.0259i$ Motivic weight: $$17$$ Character: $\chi_{8} (5, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 8,\ (\ :17/2),\ 0.999 + 0.0259i)$$

Particular Values

 $$L(9)$$ $$\approx$$ $$1.08147 - 0.0140519i$$ $$L(\frac12)$$ $$\approx$$ $$1.08147 - 0.0140519i$$ $$L(\frac{19}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (362. + 3.13i)T$$
good3 $$1 - 1.38e4iT - 1.29e8T^{2}$$
5 $$1 + 6.65e5iT - 7.62e11T^{2}$$
7 $$1 + 7.57e6T + 2.32e14T^{2}$$
11 $$1 + 1.97e8iT - 5.05e17T^{2}$$
13 $$1 + 4.74e9iT - 8.65e18T^{2}$$
17 $$1 - 3.37e10T + 8.27e20T^{2}$$
19 $$1 - 1.00e11iT - 5.48e21T^{2}$$
23 $$1 - 3.16e11T + 1.41e23T^{2}$$
29 $$1 + 3.54e12iT - 7.25e24T^{2}$$
31 $$1 - 2.76e11T + 2.25e25T^{2}$$
37 $$1 - 2.12e13iT - 4.56e26T^{2}$$
41 $$1 - 8.47e13T + 2.61e27T^{2}$$
43 $$1 + 1.38e14iT - 5.87e27T^{2}$$
47 $$1 - 8.21e13T + 2.66e28T^{2}$$
53 $$1 + 3.16e14iT - 2.05e29T^{2}$$
59 $$1 - 3.29e14iT - 1.27e30T^{2}$$
61 $$1 + 6.60e14iT - 2.24e30T^{2}$$
67 $$1 + 3.64e15iT - 1.10e31T^{2}$$
71 $$1 - 9.19e15T + 2.96e31T^{2}$$
73 $$1 + 5.44e15T + 4.74e31T^{2}$$
79 $$1 - 1.06e16T + 1.81e32T^{2}$$
83 $$1 + 9.38e15iT - 4.21e32T^{2}$$
89 $$1 - 2.24e16T + 1.37e33T^{2}$$
97 $$1 + 4.00e16T + 5.95e33T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

Imaginary part of the first few zeros on the critical line

−17.07122107663213279601170210049, −16.19331095937044013838356671630, −15.10428909378665905900397419574, −12.48651191962340683687916381779, −10.53048149139877749683420452479, −9.590720423767112409130980224019, −8.087742344461292888657965693134, −5.56416140863468238443460693757, −3.34416092327149839940736315786, −0.76062552356690262625588702070, 1.12854597882151162986182962752, 2.65632942381159082257042719693, 6.58768709989258005296661228740, 7.32754030067625590200203667942, 9.297042333426768432410034465215, 11.13583490064774578893768389328, 12.59134893637959526193557021242, 14.45916172797475659995087708821, 16.32691227338549058333272217255, 17.84046014871900871541508805088