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Properties

Label 1-33e2-1089.563-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.00576i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.272 + 0.962i)2-s + (−0.851 + 0.524i)4-s + (−0.00951 − 0.999i)5-s + (0.935 − 0.353i)7-s + (−0.736 − 0.676i)8-s + (0.959 − 0.281i)10-s + (−0.380 + 0.924i)13-s + (0.595 + 0.803i)14-s + (0.449 − 0.893i)16-s + (0.0855 − 0.996i)17-s + (0.921 − 0.389i)19-s + (0.532 + 0.846i)20-s + (−0.0475 + 0.998i)23-s + (−0.999 + 0.0190i)25-s + (−0.993 − 0.113i)26-s + ⋯
L(s)  = 1  + (0.272 + 0.962i)2-s + (−0.851 + 0.524i)4-s + (−0.00951 − 0.999i)5-s + (0.935 − 0.353i)7-s + (−0.736 − 0.676i)8-s + (0.959 − 0.281i)10-s + (−0.380 + 0.924i)13-s + (0.595 + 0.803i)14-s + (0.449 − 0.893i)16-s + (0.0855 − 0.996i)17-s + (0.921 − 0.389i)19-s + (0.532 + 0.846i)20-s + (−0.0475 + 0.998i)23-s + (−0.999 + 0.0190i)25-s + (−0.993 − 0.113i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00576i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (563, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.00576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.600693255 + 0.004617759523i\)
\(L(\frac12)\) \(\approx\) \(1.600693255 + 0.004617759523i\)
\(L(1)\) \(\approx\) \(1.164983813 + 0.2622120542i\)
\(L(1)\) \(\approx\) \(1.164983813 + 0.2622120542i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.272 + 0.962i)T \)
5 \( 1 + (-0.00951 - 0.999i)T \)
7 \( 1 + (0.935 - 0.353i)T \)
13 \( 1 + (-0.380 + 0.924i)T \)
17 \( 1 + (0.0855 - 0.996i)T \)
19 \( 1 + (0.921 - 0.389i)T \)
23 \( 1 + (-0.0475 + 0.998i)T \)
29 \( 1 + (0.483 + 0.875i)T \)
31 \( 1 + (-0.179 - 0.983i)T \)
37 \( 1 + (-0.564 - 0.825i)T \)
41 \( 1 + (-0.0665 - 0.997i)T \)
43 \( 1 + (-0.580 - 0.814i)T \)
47 \( 1 + (-0.640 + 0.768i)T \)
53 \( 1 + (0.998 + 0.0570i)T \)
59 \( 1 + (0.830 + 0.556i)T \)
61 \( 1 + (0.969 - 0.244i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (-0.974 - 0.226i)T \)
73 \( 1 + (0.254 - 0.967i)T \)
79 \( 1 + (-0.749 - 0.662i)T \)
83 \( 1 + (-0.625 - 0.780i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.00951 - 0.999i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.5240292641531128787396914011, −20.70811530115180577690855239582, −19.921909632517081461105533244804, −19.17235801197987177589003596156, −18.31185506341051626671818147523, −17.928181619132256347618843125438, −17.12536672003611149839634245973, −15.66871395961857789690166513798, −14.75052214143880211188150664030, −14.53651094441543365870290106878, −13.55455042148657901280962601838, −12.63108342699525292884671184519, −11.79256498341786625644193165428, −11.22672139874901648573045574834, −10.227114137534159377407653041374, −9.991557621326126215866339725676, −8.474030831891865546707944592822, −8.03559138991006354420096029767, −6.71834030743245183536912643337, −5.693951784416011816078282386849, −4.964702211261379561998650942695, −3.88317448703872391171757593117, −2.99165685347519387573573165634, −2.23668676483698437767533539755, −1.18247404414253480241036492466, 0.6950478602037128173081527642, 1.92406092436536252432410112337, 3.49640493943414409356562123847, 4.451852446816613978332593527791, 5.06442488228395379987548001988, 5.69454921880432094681849195641, 7.17410847546509310137896822564, 7.44395672239793006753772725027, 8.579855044465564427720583742843, 9.13963887513926267849348794837, 9.963264166282317813728962206086, 11.513337335649061054511941736156, 11.92970154197171012285587425712, 12.99905163765018817353085699887, 13.84556392782227308700174741601, 14.205959368432128405690961749236, 15.29277292195852923222616511381, 16.084962548767241921532948396679, 16.63615599192593266623807590145, 17.47671163986239385486211845369, 17.95259523099239603342541526172, 18.989137658565419394513092107628, 20.04000315878041066713259124558, 20.812765234363851681758126731650, 21.45298151285384281434744511086

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