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Properties

Label 1-2183-2183.239-r1-0-0
Degree $1$
Conductor $2183$
Sign $-0.0942 - 0.995i$
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  + (0.937 + 0.347i)2-s + (0.541 + 0.840i)3-s + (0.758 + 0.651i)4-s + (−0.614 − 0.788i)5-s + (0.214 + 0.976i)6-s + (−0.0661 − 0.997i)7-s + (0.484 + 0.874i)8-s + (−0.414 + 0.910i)9-s + (−0.302 − 0.953i)10-s + (−0.750 + 0.661i)11-s + (−0.137 + 0.990i)12-s + (0.910 − 0.414i)13-s + (0.284 − 0.958i)14-s + (0.330 − 0.943i)15-s + (0.149 + 0.988i)16-s + (0.556 − 0.831i)17-s + ⋯
L(s)  = 1  + (0.937 + 0.347i)2-s + (0.541 + 0.840i)3-s + (0.758 + 0.651i)4-s + (−0.614 − 0.788i)5-s + (0.214 + 0.976i)6-s + (−0.0661 − 0.997i)7-s + (0.484 + 0.874i)8-s + (−0.414 + 0.910i)9-s + (−0.302 − 0.953i)10-s + (−0.750 + 0.661i)11-s + (−0.137 + 0.990i)12-s + (0.910 − 0.414i)13-s + (0.284 − 0.958i)14-s + (0.330 − 0.943i)15-s + (0.149 + 0.988i)16-s + (0.556 − 0.831i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0942 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2183\)    =    \(37 \cdot 59\)
Sign: $-0.0942 - 0.995i$
Analytic conductor: \(234.595\)
Root analytic conductor: \(234.595\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2183} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2183,\ (1:\ ),\ -0.0942 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9507377064 - 1.045046039i\)
\(L(\frac12)\) \(\approx\) \(0.9507377064 - 1.045046039i\)
\(L(1)\) \(\approx\) \(1.618710425 + 0.4207973852i\)
\(L(1)\) \(\approx\) \(1.618710425 + 0.4207973852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.937 + 0.347i)T \)
3 \( 1 + (0.541 + 0.840i)T \)
5 \( 1 + (-0.614 - 0.788i)T \)
7 \( 1 + (-0.0661 - 0.997i)T \)
11 \( 1 + (-0.750 + 0.661i)T \)
13 \( 1 + (0.910 - 0.414i)T \)
17 \( 1 + (0.556 - 0.831i)T \)
19 \( 1 + (-0.919 - 0.392i)T \)
23 \( 1 + (0.576 - 0.817i)T \)
29 \( 1 + (0.179 + 0.983i)T \)
31 \( 1 + (0.998 - 0.0541i)T \)
41 \( 1 + (-0.965 + 0.261i)T \)
43 \( 1 + (-0.319 - 0.947i)T \)
47 \( 1 + (-0.999 + 0.0361i)T \)
53 \( 1 + (-0.0421 - 0.999i)T \)
61 \( 1 + (-0.999 - 0.00601i)T \)
67 \( 1 + (-0.358 + 0.933i)T \)
71 \( 1 + (-0.742 + 0.670i)T \)
73 \( 1 + (0.725 - 0.687i)T \)
79 \( 1 + (-0.678 + 0.734i)T \)
83 \( 1 + (0.313 - 0.949i)T \)
89 \( 1 + (-0.505 - 0.863i)T \)
97 \( 1 + (-0.972 - 0.232i)T \)
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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

−19.40446692811935698274083310831, −19.21151946025243644530337316377, −18.688232699086669210490656667527, −18.01394125305296333844629189903, −16.73421998058943749605785172833, −15.69124844542361343735321263875, −15.22441469890470557048787206416, −14.71406309354825965498042776650, −13.77828651671755947403636617832, −13.32989589230317923118064081998, −12.45392706905184421250866761530, −11.88019669052895276017588314556, −11.21171328273288150009710046696, −10.4978869708415521240359977260, −9.446935315965291733813651549, −8.24535148044423218740352242836, −7.99723207290612138012867477024, −6.75968767557825202010311858534, −6.20909042775857399098553360292, −5.66218948055204989183841915757, −4.34607957419278357459433173657, −3.37229846994283473308134559537, −2.99094858898330146843448740588, −2.111071931700546163070572952459, −1.262036907305495632242927063250, 0.139556489382717165928222624338, 1.5016779599004268512863929371, 2.807043013685901519213858204, 3.42674782682327278577855644302, 4.29050565653108554528226251401, 4.779329889953227417222523598426, 5.37064826888598031986831735489, 6.68231806662334467634088375911, 7.43313766524130305946268682214, 8.24232524925276047128126174393, 8.668475418273445313205976692661, 9.943879714896141454044284436204, 10.634934823250725520473144441333, 11.28232645073762074470502193457, 12.26192970302789036264506109247, 13.15341215111822054666665095963, 13.429924065531606605085311624703, 14.40157543498929005483186415615, 15.1019814542391521421447841548, 15.71326311534650456655031968574, 16.31362837046930310126556640029, 16.79443894794371474522173116321, 17.62274791643349534098071534304, 18.88653177076160434408451456353, 19.89475103163907713183624655367

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