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Properties

Label 1-229-229.28-r1-0-0
Degree $1$
Conductor $229$
Sign $-0.0938 - 0.995i$
Analytic cond. $24.6094$
Root an. cond. $24.6094$
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  + (−0.164 − 0.986i)2-s + (0.904 − 0.426i)3-s + (−0.945 + 0.324i)4-s + (0.962 + 0.272i)5-s + (−0.569 − 0.821i)6-s + (0.892 − 0.451i)7-s + (0.475 + 0.879i)8-s + (0.635 − 0.771i)9-s + (0.110 − 0.993i)10-s + (0.677 − 0.735i)11-s + (−0.716 + 0.697i)12-s + (−0.969 + 0.245i)13-s + (−0.592 − 0.805i)14-s + (0.986 − 0.164i)15-s + (0.789 − 0.614i)16-s + (0.245 + 0.969i)17-s + ⋯
L(s)  = 1  + (−0.164 − 0.986i)2-s + (0.904 − 0.426i)3-s + (−0.945 + 0.324i)4-s + (0.962 + 0.272i)5-s + (−0.569 − 0.821i)6-s + (0.892 − 0.451i)7-s + (0.475 + 0.879i)8-s + (0.635 − 0.771i)9-s + (0.110 − 0.993i)10-s + (0.677 − 0.735i)11-s + (−0.716 + 0.697i)12-s + (−0.969 + 0.245i)13-s + (−0.592 − 0.805i)14-s + (0.986 − 0.164i)15-s + (0.789 − 0.614i)16-s + (0.245 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(229\)
Sign: $-0.0938 - 0.995i$
Analytic conductor: \(24.6094\)
Root analytic conductor: \(24.6094\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{229} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 229,\ (1:\ ),\ -0.0938 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.066726551 - 2.270649601i\)
\(L(\frac12)\) \(\approx\) \(2.066726551 - 2.270649601i\)
\(L(1)\) \(\approx\) \(1.389317553 - 0.9386219856i\)
\(L(1)\) \(\approx\) \(1.389317553 - 0.9386219856i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad229 \( 1 \)
good2 \( 1 + (-0.164 - 0.986i)T \)
3 \( 1 + (0.904 - 0.426i)T \)
5 \( 1 + (0.962 + 0.272i)T \)
7 \( 1 + (0.892 - 0.451i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (-0.969 + 0.245i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (0.716 + 0.697i)T \)
23 \( 1 + (0.110 + 0.993i)T \)
29 \( 1 + (0.0550 - 0.998i)T \)
31 \( 1 + (0.954 - 0.298i)T \)
37 \( 1 + (0.137 - 0.990i)T \)
41 \( 1 + (-0.936 - 0.350i)T \)
43 \( 1 + (0.789 + 0.614i)T \)
47 \( 1 + (-0.771 - 0.635i)T \)
53 \( 1 + (-0.0825 + 0.996i)T \)
59 \( 1 + (-0.990 + 0.137i)T \)
61 \( 1 + (-0.986 - 0.164i)T \)
67 \( 1 + (0.936 - 0.350i)T \)
71 \( 1 + (-0.975 + 0.218i)T \)
73 \( 1 + (-0.523 + 0.851i)T \)
79 \( 1 + (0.0550 + 0.998i)T \)
83 \( 1 + (-0.926 - 0.376i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (-0.0275 - 0.999i)T \)
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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

−26.220504800488003659500700282163, −25.269155652705120085587191977311, −24.788085496875312683160746423405, −24.168657520887510504631316454, −22.44385985984582179125547319293, −21.91933198288496938197847596017, −20.779016813648788775593515115694, −19.94088473466127075731188119874, −18.61186654697213371921885067067, −17.77540926219144631836676890866, −16.95350433947279150615237876538, −15.855597151722549053762289881930, −14.7634481601614091723610516195, −14.34880042787008604378085003311, −13.42282067221727375369796476744, −12.18658753946485648289410919417, −10.31563456789801495183149833674, −9.46714035406803315413535336183, −8.8296315298676298402100035460, −7.72033008566617360790793812294, −6.6908088953355658764117050424, −4.973506574217447133729018453113, −4.79398391214056934146626014002, −2.76593746359952080703777837675, −1.39094610433869018479068389387, 1.18883758478596203980233917070, 1.95063438603632249789535557907, 3.16315279933920026447207850630, 4.2884793696309933652082722734, 5.80995397157223178017959864443, 7.410232550797897159728553258031, 8.35044189545165938738088559455, 9.43922680149128302044449862617, 10.1517205758459008324786363997, 11.39760261458833098222361329865, 12.40982674767142961757737403838, 13.64809287037718335344361368607, 14.05181752799585560976877284796, 14.87490648389870658910824789727, 17.003862321231597637580405287526, 17.55015931516154030704716257034, 18.61954421130456006972183416002, 19.385965015692218974884427160411, 20.248721851292541759073654782677, 21.339339303848991283540265268721, 21.57409173655779724439842443668, 22.96923204949194828441501422694, 24.25931878168581195370898564099, 24.92094662747869457474980140272, 26.227936190621035027380489061211

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