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Properties

Label 2-2e3-8.3-c4-0-0
Degree $2$
Conductor $8$
Sign $0.687 - 0.726i$
Analytic cond. $0.826959$
Root an. cond. $0.909373$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 3.87i)2-s + 6·3-s + (−14.0 − 7.74i)4-s − 30.9i·5-s + (−6 + 23.2i)6-s + 61.9i·7-s + (44.0 − 46.4i)8-s − 45·9-s + (120. + 30.9i)10-s − 26·11-s + (−84.0 − 46.4i)12-s + 30.9i·13-s + (−240. − 61.9i)14-s − 185. i·15-s + (136. + 216. i)16-s + 226·17-s + ⋯
L(s)  = 1  + (−0.250 + 0.968i)2-s + 0.666·3-s + (−0.875 − 0.484i)4-s − 1.23i·5-s + (−0.166 + 0.645i)6-s + 1.26i·7-s + (0.687 − 0.726i)8-s − 0.555·9-s + (1.20 + 0.309i)10-s − 0.214·11-s + (−0.583 − 0.322i)12-s + 0.183i·13-s + (−1.22 − 0.316i)14-s − 0.826i·15-s + (0.531 + 0.847i)16-s + 0.782·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.687 - 0.726i$
Analytic conductor: \(0.826959\)
Root analytic conductor: \(0.909373\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :2),\ 0.687 - 0.726i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.855528 + 0.368160i\)
\(L(\frac12)\) \(\approx\) \(0.855528 + 0.368160i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 3.87i)T \)
good3 \( 1 - 6T + 81T^{2} \)
5 \( 1 + 30.9iT - 625T^{2} \)
7 \( 1 - 61.9iT - 2.40e3T^{2} \)
11 \( 1 + 26T + 1.46e4T^{2} \)
13 \( 1 - 30.9iT - 2.85e4T^{2} \)
17 \( 1 - 226T + 8.35e4T^{2} \)
19 \( 1 - 134T + 1.30e5T^{2} \)
23 \( 1 + 309. iT - 2.79e5T^{2} \)
29 \( 1 + 340. iT - 7.07e5T^{2} \)
31 \( 1 - 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.76e3iT - 1.87e6T^{2} \)
41 \( 1 - 994T + 2.82e6T^{2} \)
43 \( 1 + 1.88e3T + 3.41e6T^{2} \)
47 \( 1 + 2.10e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.81e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.01e3T + 1.21e7T^{2} \)
61 \( 1 + 2.07e3iT - 1.38e7T^{2} \)
67 \( 1 - 8.00e3T + 2.01e7T^{2} \)
71 \( 1 - 557. iT - 2.54e7T^{2} \)
73 \( 1 - 386T + 2.83e7T^{2} \)
79 \( 1 - 1.10e4iT - 3.89e7T^{2} \)
83 \( 1 + 2.23e3T + 4.74e7T^{2} \)
89 \( 1 + 1.00e4T + 6.27e7T^{2} \)
97 \( 1 - 8.73e3T + 8.85e7T^{2} \)
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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

−21.41974751600567597501507952751, −19.88017054250292570081466802057, −18.47094474270830168394216425506, −16.84746508040984966204166572848, −15.59975009337965161486346141718, −14.18644411469039162318032379072, −12.49505108077315281617686365600, −9.223835836731734465687979184813, −8.300859782990254159011153936074, −5.42938113619192888783297542980, 3.27029599977862769934910771922, 7.73365253042416185622777892779, 9.965584594603764605037194581979, 11.25256454262619031776370893028, 13.54086326319204164680456829676, 14.54604406493841985013454267836, 17.11849225917962360839081028424, 18.57549211362932464980201114502, 19.75554523375872504894024081820, 20.75851879792913376932067769058

Graph of the $Z$-function along the critical line