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Properties

Label 1-5160-5160.1733-r0-0-0
Degree $1$
Conductor $5160$
Sign $0.959 - 0.281i$
Analytic cond. $23.9629$
Root an. cond. $23.9629$
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  + (−0.866 + 0.5i)7-s + (0.623 − 0.781i)11-s + (0.680 + 0.733i)13-s + (0.294 + 0.955i)17-s + (−0.988 + 0.149i)19-s + (0.930 − 0.365i)23-s + (−0.0747 − 0.997i)29-s + (0.826 − 0.563i)31-s + (−0.866 − 0.5i)37-s + (0.900 + 0.433i)41-s + (−0.781 + 0.623i)47-s + (0.5 − 0.866i)49-s + (0.680 − 0.733i)53-s + (0.222 − 0.974i)59-s + (−0.826 − 0.563i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)7-s + (0.623 − 0.781i)11-s + (0.680 + 0.733i)13-s + (0.294 + 0.955i)17-s + (−0.988 + 0.149i)19-s + (0.930 − 0.365i)23-s + (−0.0747 − 0.997i)29-s + (0.826 − 0.563i)31-s + (−0.866 − 0.5i)37-s + (0.900 + 0.433i)41-s + (−0.781 + 0.623i)47-s + (0.5 − 0.866i)49-s + (0.680 − 0.733i)53-s + (0.222 − 0.974i)59-s + (−0.826 − 0.563i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5160\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 43\)
Sign: $0.959 - 0.281i$
Analytic conductor: \(23.9629\)
Root analytic conductor: \(23.9629\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5160} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5160,\ (0:\ ),\ 0.959 - 0.281i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.576744995 - 0.2263764765i\)
\(L(\frac12)\) \(\approx\) \(1.576744995 - 0.2263764765i\)
\(L(1)\) \(\approx\) \(1.041172920 + 0.008799364828i\)
\(L(1)\) \(\approx\) \(1.041172920 + 0.008799364828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (0.680 + 0.733i)T \)
17 \( 1 + (0.294 + 0.955i)T \)
19 \( 1 + (-0.988 + 0.149i)T \)
23 \( 1 + (0.930 - 0.365i)T \)
29 \( 1 + (-0.0747 - 0.997i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (-0.781 + 0.623i)T \)
53 \( 1 + (0.680 - 0.733i)T \)
59 \( 1 + (0.222 - 0.974i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (-0.149 - 0.988i)T \)
71 \( 1 + (-0.365 + 0.930i)T \)
73 \( 1 + (-0.680 - 0.733i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.997 + 0.0747i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (-0.781 - 0.623i)T \)
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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

−17.83752494085041092604294681600, −17.47695830731108845219670898882, −16.57585923894358685037575529899, −16.18292186057664445873506770879, −15.268722567286216531286430515230, −14.889631292055281678635885381327, −13.85635654227221689643545096637, −13.43036790192195947002682891602, −12.647740146651493724350747016401, −12.168447300615543966714505871936, −11.279731895258287428437414979261, −10.45748096800910292682473465344, −10.10175156177088606062282137330, −9.10404635146886891479389196594, −8.7800137767639616680698139087, −7.64629380441310055758714580353, −7.0173402623506039657978702946, −6.53181242716352494940691219831, −5.65581264831530901110155333684, −4.829041932867249760715939363717, −4.08189237696415665039647389276, −3.27813171802918953473611397922, −2.71933557323196472629671836887, −1.517567974988124498469081877, −0.77182169620322504637003212907, 0.57487860857764586324005899074, 1.62816499656941692409281835693, 2.44501372351252085085431161941, 3.38353238383801961242980985854, 3.89827017638149942007684684547, 4.72394590910852978954317345692, 5.888712080949420453236573793709, 6.248799216984273711221932689409, 6.740869235822400136426521435861, 7.89875683679934817358192463624, 8.60132929216420226536460187499, 9.07421365099972332147081951537, 9.80777815007350422172493516947, 10.616734054361567252176382360217, 11.281333868504535609362438266476, 11.920704744585973574332297211493, 12.75483377161623100528160548677, 13.16853592581097183208547052879, 13.998349979877417550973638520671, 14.64711730450890475456667612040, 15.376969885214909791345126023144, 15.96842767822005605824178868064, 16.81489025880341999678853217225, 16.98810254320008341368600025778, 18.025942488207184182086104651332

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