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Properties

Label 1-2183-2183.133-r1-0-0
Degree $1$
Conductor $2183$
Sign $-0.955 + 0.295i$
Analytic cond. $234.595$
Root an. cond. $234.595$
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  + (0.463 − 0.886i)2-s + (−0.510 − 0.859i)3-s + (−0.571 − 0.820i)4-s + (−0.813 − 0.580i)5-s + (−0.998 + 0.0541i)6-s + (0.917 − 0.397i)7-s + (−0.992 + 0.126i)8-s + (−0.479 + 0.877i)9-s + (−0.891 + 0.452i)10-s + (0.983 − 0.179i)11-s + (−0.414 + 0.910i)12-s + (0.877 − 0.479i)13-s + (0.0721 − 0.997i)14-s + (−0.0841 + 0.996i)15-s + (−0.347 + 0.937i)16-s + (−0.595 − 0.803i)17-s + ⋯
L(s)  = 1  + (0.463 − 0.886i)2-s + (−0.510 − 0.859i)3-s + (−0.571 − 0.820i)4-s + (−0.813 − 0.580i)5-s + (−0.998 + 0.0541i)6-s + (0.917 − 0.397i)7-s + (−0.992 + 0.126i)8-s + (−0.479 + 0.877i)9-s + (−0.891 + 0.452i)10-s + (0.983 − 0.179i)11-s + (−0.414 + 0.910i)12-s + (0.877 − 0.479i)13-s + (0.0721 − 0.997i)14-s + (−0.0841 + 0.996i)15-s + (−0.347 + 0.937i)16-s + (−0.595 − 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $-0.955 + 0.295i$
Analytic conductor: \(234.595\)
Root analytic conductor: \(234.595\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (1:\ ),\ -0.955 + 0.295i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3751356414 - 2.482317673i\)
\(L(\frac12)\) \(\approx\) \(-0.3751356414 - 2.482317673i\)
\(L(1)\) \(\approx\) \(0.6219552886 - 1.009452743i\)
\(L(1)\) \(\approx\) \(0.6219552886 - 1.009452743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.463 - 0.886i)T \)
3 \( 1 + (-0.510 - 0.859i)T \)
5 \( 1 + (-0.813 - 0.580i)T \)
7 \( 1 + (0.917 - 0.397i)T \)
11 \( 1 + (0.983 - 0.179i)T \)
13 \( 1 + (0.877 - 0.479i)T \)
17 \( 1 + (-0.595 - 0.803i)T \)
19 \( 1 + (0.473 - 0.880i)T \)
23 \( 1 + (-0.806 + 0.590i)T \)
29 \( 1 + (0.738 + 0.674i)T \)
31 \( 1 + (0.928 + 0.370i)T \)
41 \( 1 + (0.0661 + 0.997i)T \)
43 \( 1 + (0.762 + 0.647i)T \)
47 \( 1 + (0.700 - 0.713i)T \)
53 \( 1 + (-0.392 - 0.919i)T \)
61 \( 1 + (0.924 - 0.381i)T \)
67 \( 1 + (0.955 - 0.296i)T \)
71 \( 1 + (0.824 + 0.566i)T \)
73 \( 1 + (0.561 - 0.827i)T \)
79 \( 1 + (-0.982 - 0.185i)T \)
83 \( 1 + (0.951 - 0.307i)T \)
89 \( 1 + (0.132 + 0.991i)T \)
97 \( 1 + (-0.899 - 0.436i)T \)
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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

−20.236257373196044260892303897089, −19.01432572486055590642490059936, −18.35618542509398740985580027566, −17.494876566708783431126982556310, −17.10070748281601880252554832585, −16.05815423445258908271511307015, −15.65612374440669384508541044750, −15.05097970368109494620496771903, −14.24423618398983065631836263624, −13.9983058151503764923291164237, −12.362760428305637854580238643657, −11.98191157545606295229108326135, −11.31486159742328375457964694433, −10.5575025292257808875421120872, −9.50349113762870258303735673583, −8.54918501385462394292197756118, −8.21688872419568592856631303748, −7.1011561137103494051351855604, −6.22318096584527270820216636429, −5.857235235477855174763073940693, −4.616526352039746718245567853840, −4.09493810921520515927361459929, −3.686161650580462657044735485537, −2.387535983344324316508102451255, −0.82830726684020420244955331232, 0.647287588577654054413858808314, 0.96018512580318869025180460568, 1.81784140741264695487007007968, 2.979573266282133722903535612572, 3.9274314489236628204711191351, 4.74137699610391501197397750826, 5.26962247487947749551888447247, 6.33086457419491235461469889467, 7.11377749981959899914242769244, 8.16934713271740802929530382791, 8.63358106378977641215259193831, 9.64127673037030085871259455169, 10.89887643162188471537904710639, 11.28041273990875386158829153439, 11.799597424054937587830635250573, 12.41228302877585287170977056736, 13.36664176126391830578142407093, 13.77901278936679085475771044531, 14.52007018410468265391779759028, 15.59964756115519771293716020370, 16.227843015357238077248727948058, 17.40672151366452296792698750464, 17.798423264854565508227925129197, 18.49171774099099371937595724334, 19.428786763521919579147243793960

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