↑

Properties

Label 2-2e3-8.5-c5-0-3
Degree $2$
Conductor $8$
Sign $-0.129 + 0.991i$
Analytic cond. $1.28307$
Root an. cond. $1.13272$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.77 − 3.03i)2-s − 23.6i·3-s + (13.5 + 28.9i)4-s + 1.38i·5-s + (−71.7 + 112. i)6-s + 160.·7-s + (23.4 − 179. i)8-s − 314.·9-s + (4.20 − 6.61i)10-s − 129. i·11-s + (684. − 319. i)12-s + 759. i·13-s + (−766. − 488. i)14-s + 32.7·15-s + (−657. + 785. i)16-s + 323.·17-s + ⋯
L(s)  = 1  + (−0.843 − 0.537i)2-s − 1.51i·3-s + (0.423 + 0.906i)4-s + 0.0247i·5-s + (−0.813 + 1.27i)6-s + 1.23·7-s + (0.129 − 0.991i)8-s − 1.29·9-s + (0.0133 − 0.0209i)10-s − 0.321i·11-s + (1.37 − 0.641i)12-s + 1.24i·13-s + (−1.04 − 0.665i)14-s + 0.0375·15-s + (−0.641 + 0.766i)16-s + 0.271·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.129 + 0.991i$
Analytic conductor: \(1.28307\)
Root analytic conductor: \(1.13272\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :5/2),\ -0.129 + 0.991i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.542464 - 0.617909i\)
\(L(\frac12)\) \(\approx\) \(0.542464 - 0.617909i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.77 + 3.03i)T \)
good3 \( 1 + 23.6iT - 243T^{2} \)
5 \( 1 - 1.38iT - 3.12e3T^{2} \)
7 \( 1 - 160.T + 1.68e4T^{2} \)
11 \( 1 + 129. iT - 1.61e5T^{2} \)
13 \( 1 - 759. iT - 3.71e5T^{2} \)
17 \( 1 - 323.T + 1.41e6T^{2} \)
19 \( 1 - 198. iT - 2.47e6T^{2} \)
23 \( 1 + 1.19e3T + 6.43e6T^{2} \)
29 \( 1 - 5.98e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.87e3T + 2.86e7T^{2} \)
37 \( 1 + 3.69e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.04e4T + 1.15e8T^{2} \)
43 \( 1 + 9.87e3iT - 1.47e8T^{2} \)
47 \( 1 + 6.29e3T + 2.29e8T^{2} \)
53 \( 1 - 2.17e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.35e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.85e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.31e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.94e4T + 1.80e9T^{2} \)
73 \( 1 - 5.12e4T + 2.07e9T^{2} \)
79 \( 1 + 7.37e4T + 3.07e9T^{2} \)
83 \( 1 + 6.16e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.06e5T + 5.58e9T^{2} \)
97 \( 1 - 1.25e4T + 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.19623589684634909554941707895, −18.79728680469505610932950690535, −18.12311756650053989879601767377, −16.78712241271852391835648136341, −14.10746190163334342025484236232, −12.40471134696526915104462435550, −11.21040594373596566035310849962, −8.557130676955492868920538131225, −7.12561003504052907701598768524, −1.63932226537607778906786474852, 5.11224965310140540215494532693, 8.205562942966769425332630197734, 9.912456974173941537052175492134, 11.11460151679582914023082211631, 14.62218622970180512937539015126, 15.46159282788681824118615799503, 16.86910616004545062190342775146, 18.05799950204718519695155937959, 20.15597759506445499112301233918, 21.02535708629812350169819063177

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