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Properties

Label 1-2183-2183.241-r1-0-0
Degree $1$
Conductor $2183$
Sign $0.297 + 0.954i$
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  + (0.889 + 0.457i)2-s + (−0.951 + 0.307i)3-s + (0.580 + 0.813i)4-s + (0.993 − 0.114i)5-s + (−0.986 − 0.161i)6-s + (0.985 − 0.167i)7-s + (0.143 + 0.989i)8-s + (0.810 − 0.585i)9-s + (0.935 + 0.353i)10-s + (0.0180 − 0.999i)11-s + (−0.803 − 0.595i)12-s + (−0.585 + 0.810i)13-s + (0.953 + 0.302i)14-s + (−0.910 + 0.414i)15-s + (−0.324 + 0.945i)16-s + (0.937 − 0.347i)17-s + ⋯
L(s)  = 1  + (0.889 + 0.457i)2-s + (−0.951 + 0.307i)3-s + (0.580 + 0.813i)4-s + (0.993 − 0.114i)5-s + (−0.986 − 0.161i)6-s + (0.985 − 0.167i)7-s + (0.143 + 0.989i)8-s + (0.810 − 0.585i)9-s + (0.935 + 0.353i)10-s + (0.0180 − 0.999i)11-s + (−0.803 − 0.595i)12-s + (−0.585 + 0.810i)13-s + (0.953 + 0.302i)14-s + (−0.910 + 0.414i)15-s + (−0.324 + 0.945i)16-s + (0.937 − 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $0.297 + 0.954i$
Analytic conductor: \(234.595\)
Root analytic conductor: \(234.595\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (1:\ ),\ 0.297 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.017766390 + 2.955444188i\)
\(L(\frac12)\) \(\approx\) \(4.017766390 + 2.955444188i\)
\(L(1)\) \(\approx\) \(1.842117209 + 0.7715770359i\)
\(L(1)\) \(\approx\) \(1.842117209 + 0.7715770359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.889 + 0.457i)T \)
3 \( 1 + (-0.951 + 0.307i)T \)
5 \( 1 + (0.993 - 0.114i)T \)
7 \( 1 + (0.985 - 0.167i)T \)
11 \( 1 + (0.0180 - 0.999i)T \)
13 \( 1 + (-0.585 + 0.810i)T \)
17 \( 1 + (0.937 - 0.347i)T \)
19 \( 1 + (0.966 - 0.255i)T \)
23 \( 1 + (0.661 + 0.750i)T \)
29 \( 1 + (-0.386 + 0.922i)T \)
31 \( 1 + (0.419 - 0.907i)T \)
41 \( 1 + (-0.781 + 0.624i)T \)
43 \( 1 + (-0.515 - 0.856i)T \)
47 \( 1 + (-0.232 - 0.972i)T \)
53 \( 1 + (0.510 + 0.859i)T \)
61 \( 1 + (-0.296 + 0.955i)T \)
67 \( 1 + (0.880 - 0.473i)T \)
71 \( 1 + (0.551 + 0.834i)T \)
73 \( 1 + (-0.976 + 0.214i)T \)
79 \( 1 + (0.375 + 0.926i)T \)
83 \( 1 + (0.971 + 0.238i)T \)
89 \( 1 + (-0.975 - 0.220i)T \)
97 \( 1 + (-0.738 - 0.674i)T \)
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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

−19.44919993584953879342356012943, −18.566297783826787717056364844672, −17.99284571549556658756563597151, −17.35585334277678333470245660257, −16.70657004364719651312907008839, −15.64837200084500749548920989262, −14.840340407681051563558087659104, −14.34857854195094025628724248102, −13.47763901628409407000671842317, −12.709954939943357055654179744951, −12.22008540570457888344944711924, −11.54156478025454894686109821820, −10.62435838257147223929792812447, −10.15538120424487135045318566386, −9.517042511623013729120482679514, −7.98031242353981544292908959029, −7.222099683113110536840202872907, −6.422082652123644340167126201174, −5.57245978130523279654931006252, −5.08742518913894133095835551358, −4.58165415001953130550238603359, −3.21190218493342094908972079398, −2.20501189648762509562143169816, −1.56755931384844016953186955949, −0.79663312736214188609914871694, 0.94789046816281916786982192794, 1.765091096935245289584156188270, 2.957879116564622522953457935965, 3.88903452443301288434057721347, 4.942279552171208209367788732902, 5.29196092951627619344451281116, 5.8609944512514535700038156332, 6.874498012285193727554861815972, 7.39694150096675756290350225146, 8.53716550830487710043593269090, 9.43248872480798836612519154807, 10.29400848645091582473221102632, 11.2551719076739202806253435489, 11.618206127430386499525797374689, 12.358682568272778625873397752630, 13.42781618433199060016685105500, 13.8343536089228619572340239906, 14.598879350296252794815265544363, 15.30183110571291536486565338759, 16.31509192395819527633676837087, 16.886652821764938189989235681016, 17.11983784112917851313486735621, 18.132951330791638547268186204816, 18.64284098103073401071373452957, 20.065186145340596929681253501658

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