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Properties

Label 1-2183-2183.198-r1-0-0
Degree $1$
Conductor $2183$
Sign $0.157 + 0.987i$
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  + (−0.990 + 0.137i)2-s + (−0.917 − 0.397i)3-s + (0.961 − 0.273i)4-s + (0.924 + 0.381i)5-s + (0.963 + 0.267i)6-s + (0.734 − 0.678i)7-s + (−0.915 + 0.403i)8-s + (0.683 + 0.730i)9-s + (−0.968 − 0.250i)10-s + (−0.989 + 0.143i)11-s + (−0.991 − 0.132i)12-s + (0.730 + 0.683i)13-s + (−0.633 + 0.773i)14-s + (−0.696 − 0.717i)15-s + (0.850 − 0.525i)16-s + (0.975 + 0.220i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.137i)2-s + (−0.917 − 0.397i)3-s + (0.961 − 0.273i)4-s + (0.924 + 0.381i)5-s + (0.963 + 0.267i)6-s + (0.734 − 0.678i)7-s + (−0.915 + 0.403i)8-s + (0.683 + 0.730i)9-s + (−0.968 − 0.250i)10-s + (−0.989 + 0.143i)11-s + (−0.991 − 0.132i)12-s + (0.730 + 0.683i)13-s + (−0.633 + 0.773i)14-s + (−0.696 − 0.717i)15-s + (0.850 − 0.525i)16-s + (0.975 + 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.157 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $0.157 + 0.987i$
Analytic conductor: \(234.595\)
Root analytic conductor: \(234.595\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (198, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (1:\ ),\ 0.157 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178690069 + 1.005121034i\)
\(L(\frac12)\) \(\approx\) \(1.178690069 + 1.005121034i\)
\(L(1)\) \(\approx\) \(0.7622941422 + 0.1325080831i\)
\(L(1)\) \(\approx\) \(0.7622941422 + 0.1325080831i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.990 + 0.137i)T \)
3 \( 1 + (-0.917 - 0.397i)T \)
5 \( 1 + (0.924 + 0.381i)T \)
7 \( 1 + (0.734 - 0.678i)T \)
11 \( 1 + (-0.989 + 0.143i)T \)
13 \( 1 + (0.730 + 0.683i)T \)
17 \( 1 + (0.975 + 0.220i)T \)
19 \( 1 + (0.0240 + 0.999i)T \)
23 \( 1 + (0.484 + 0.874i)T \)
29 \( 1 + (0.0361 + 0.999i)T \)
31 \( 1 + (0.319 + 0.947i)T \)
41 \( 1 + (0.987 + 0.155i)T \)
43 \( 1 + (0.928 - 0.370i)T \)
47 \( 1 + (-0.302 + 0.953i)T \)
53 \( 1 + (0.995 + 0.0961i)T \)
61 \( 1 + (0.977 + 0.209i)T \)
67 \( 1 + (0.279 - 0.960i)T \)
71 \( 1 + (-0.844 + 0.536i)T \)
73 \( 1 + (0.161 - 0.986i)T \)
79 \( 1 + (-0.834 + 0.551i)T \)
83 \( 1 + (0.985 + 0.167i)T \)
89 \( 1 + (0.307 - 0.951i)T \)
97 \( 1 + (-0.353 + 0.935i)T \)
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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

−19.049783744817622236289134359926, −18.451369123246700891805349087505, −17.85703282258742666492441954729, −17.47234399012858958210450279161, −16.6674094244449501195712270854, −15.99269576473188004489306489436, −15.384347112602823908564813938873, −14.62800266124262703804908074518, −13.28286203267971036340684567536, −12.718622471686834383090294149878, −11.84765175273738059931609280704, −11.16336952240148948409243345642, −10.51382750239205406699421954979, −9.91237058918867594312084910824, −9.13020784721200021156217646959, −8.40504570951394732972785578910, −7.63353097708897533193463684972, −6.550821064188415369341783671861, −5.66388945496025271199502712755, −5.43545785239220296873514378318, −4.32302411733462570353562786530, −2.88031003806429598894581365032, −2.23930239866837075454591852757, −1.012361536884104743329785883632, −0.52978217890831023881820569662, 1.069313496881860369580273032102, 1.4097582933096190666622683148, 2.31035267925009531093303209481, 3.52341462042807485705436741365, 4.9505801831113098918431518184, 5.62907333013334552857270541551, 6.25241873219172168355750765376, 7.20300378586825686090665192051, 7.60017118519514654742550842006, 8.517975197897390364881494111547, 9.57339604431583439622236233519, 10.39992557378444474912756547254, 10.67713444761178571408996352478, 11.38602239245212985584573676615, 12.28004006643770959100473306181, 13.1149899491390685770783728014, 14.047285818884817924219197600341, 14.5844638204207929543570872571, 15.76913913645900852977226508108, 16.419252788801423083705094708375, 16.996042456127859098566744685206, 17.74220838567856290953028391455, 18.09238107863253797539107303758, 18.75216941083081881379575032800, 19.36965636143294353270460724848

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