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Properties

Label 1-2183-2183.2145-r0-0-0
Degree $1$
Conductor $2183$
Sign $0.513 + 0.857i$
Analytic cond. $10.1378$
Root an. cond. $10.1378$
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  + (−0.468 − 0.883i)2-s + (−0.725 − 0.687i)3-s + (−0.561 + 0.827i)4-s + (−0.976 − 0.214i)5-s + (−0.267 + 0.963i)6-s + (0.796 + 0.605i)7-s + (0.994 + 0.108i)8-s + (0.0541 + 0.998i)9-s + (0.267 + 0.963i)10-s + (−0.370 + 0.928i)11-s + (0.976 − 0.214i)12-s + (−0.0541 + 0.998i)13-s + (0.161 − 0.986i)14-s + (0.561 + 0.827i)15-s + (−0.370 − 0.928i)16-s + (−0.796 + 0.605i)17-s + ⋯
L(s)  = 1  + (−0.468 − 0.883i)2-s + (−0.725 − 0.687i)3-s + (−0.561 + 0.827i)4-s + (−0.976 − 0.214i)5-s + (−0.267 + 0.963i)6-s + (0.796 + 0.605i)7-s + (0.994 + 0.108i)8-s + (0.0541 + 0.998i)9-s + (0.267 + 0.963i)10-s + (−0.370 + 0.928i)11-s + (0.976 − 0.214i)12-s + (−0.0541 + 0.998i)13-s + (0.161 − 0.986i)14-s + (0.561 + 0.827i)15-s + (−0.370 − 0.928i)16-s + (−0.796 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $0.513 + 0.857i$
Analytic conductor: \(10.1378\)
Root analytic conductor: \(10.1378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (2145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (0:\ ),\ 0.513 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5545861194 + 0.3142644635i\)
\(L(\frac12)\) \(\approx\) \(0.5545861194 + 0.3142644635i\)
\(L(1)\) \(\approx\) \(0.5728801502 - 0.1079842638i\)
\(L(1)\) \(\approx\) \(0.5728801502 - 0.1079842638i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.468 - 0.883i)T \)
3 \( 1 + (-0.725 - 0.687i)T \)
5 \( 1 + (-0.976 - 0.214i)T \)
7 \( 1 + (0.796 + 0.605i)T \)
11 \( 1 + (-0.370 + 0.928i)T \)
13 \( 1 + (-0.0541 + 0.998i)T \)
17 \( 1 + (-0.796 + 0.605i)T \)
19 \( 1 + (0.947 + 0.319i)T \)
23 \( 1 + (0.856 + 0.515i)T \)
29 \( 1 + (-0.468 + 0.883i)T \)
31 \( 1 + (0.947 - 0.319i)T \)
41 \( 1 + (-0.856 + 0.515i)T \)
43 \( 1 + (0.370 + 0.928i)T \)
47 \( 1 + (0.976 - 0.214i)T \)
53 \( 1 + (0.267 - 0.963i)T \)
61 \( 1 + (-0.468 - 0.883i)T \)
67 \( 1 + (-0.994 - 0.108i)T \)
71 \( 1 + (0.976 - 0.214i)T \)
73 \( 1 + (-0.161 + 0.986i)T \)
79 \( 1 + (0.725 - 0.687i)T \)
83 \( 1 + (0.647 + 0.762i)T \)
89 \( 1 + (-0.468 + 0.883i)T \)
97 \( 1 + (0.161 + 0.986i)T \)
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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

−19.57424212356115934877262618760, −18.58518139705558902176016336217, −18.117953501765960570960688136871, −17.27486353259215671137680351054, −16.808566268642108874181785823473, −15.898554596664392518638217084728, −15.49785863456948739584598927904, −14.97999990515052432248937574611, −13.98144392577179556705779291957, −13.355615451537290898503983437271, −12.068533964698277588154962479170, −11.29535755174911915788994315364, −10.713206731548233309578172325090, −10.27178734285923294630850054620, −9.069797767835089395935966127075, −8.47277164566787435033082409409, −7.57700285905781921139185761142, −7.08677830961965890509073736477, −6.070737335697716070223754774877, −5.21481587298982038905496241742, −4.68534103705568700746086403086, −3.87730825151838056309193251052, −2.861152321662559739637192825632, −0.93125579567349387772230814047, −0.41439485111626520763038626917, 1.13120815629154647678418362722, 1.78724372070214652866260767192, 2.63357849185557582300239235109, 3.85855978878457331631239921311, 4.77844100824602164028535672748, 5.12786798142903176360639926109, 6.607315164403507913328856660390, 7.42006353825592213548844084319, 7.94175337022029856841675775492, 8.7307997281371669744008106743, 9.52130455519138621837540177685, 10.6408995547721795259442521599, 11.277224750418316901468667081432, 11.76202717303873444603233967868, 12.31106821087111928926561217438, 12.95953478183179292112349798249, 13.78741807449679531853861259762, 14.8392812134160925591408035520, 15.67579261247967876228937237332, 16.525073118686801890565447065127, 17.21411239192526951647545495491, 17.90675698213148912292866106594, 18.48281086501411015802057263421, 19.062197099390542284959024515137, 19.750178782491595242888038596046

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