Properties

Label 1-837-837.157-r0-0-0
Degree $1$
Conductor $837$
Sign $0.537 - 0.843i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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 − 0.139i)2-s + (0.961 − 0.275i)4-s + (0.766 − 0.642i)5-s + (0.559 − 0.829i)7-s + (0.913 − 0.406i)8-s + (0.669 − 0.743i)10-s + (−0.997 − 0.0697i)11-s + (0.990 + 0.139i)13-s + (0.438 − 0.898i)14-s + (0.848 − 0.529i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.559 − 0.829i)20-s + (−0.997 + 0.0697i)22-s + (0.559 + 0.829i)23-s + ⋯
L(s)  = 1  + (0.990 − 0.139i)2-s + (0.961 − 0.275i)4-s + (0.766 − 0.642i)5-s + (0.559 − 0.829i)7-s + (0.913 − 0.406i)8-s + (0.669 − 0.743i)10-s + (−0.997 − 0.0697i)11-s + (0.990 + 0.139i)13-s + (0.438 − 0.898i)14-s + (0.848 − 0.529i)16-s + (−0.104 + 0.994i)17-s + (−0.978 − 0.207i)19-s + (0.559 − 0.829i)20-s + (−0.997 + 0.0697i)22-s + (0.559 + 0.829i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.537 - 0.843i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.537 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.004284231 - 1.648743209i\)
\(L(\frac12)\) \(\approx\) \(3.004284231 - 1.648743209i\)
\(L(1)\) \(\approx\) \(2.170074711 - 0.6651082660i\)
\(L(1)\) \(\approx\) \(2.170074711 - 0.6651082660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.990 - 0.139i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.559 - 0.829i)T \)
11 \( 1 + (-0.997 - 0.0697i)T \)
13 \( 1 + (0.990 + 0.139i)T \)
17 \( 1 + (-0.104 + 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.559 + 0.829i)T \)
29 \( 1 + (0.990 - 0.139i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.882 + 0.469i)T \)
43 \( 1 + (-0.374 + 0.927i)T \)
47 \( 1 + (0.0348 + 0.999i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.374 - 0.927i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (-0.719 + 0.694i)T \)
83 \( 1 + (0.990 - 0.139i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (-0.997 - 0.0697i)T \)
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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

−22.24776924578494334260644526690, −21.4446048203254286037676677950, −20.96758941439505963781485466975, −20.33000692817398531095007929416, −18.81691611694620803983373981493, −18.40770829403137786422308069032, −17.46591809066807753700555555944, −16.49384637085971292310131994690, −15.41925355858987824229317802521, −15.13682372727030666459411615733, −14.020392288808007347440298950217, −13.55595927781971604339710174563, −12.62848971815751434786855457337, −11.78331892945638420995718917548, −10.78490697036937706942050685913, −10.37173330974258193121528646179, −8.88061998140287306805277868793, −8.07168242161255791202771474559, −6.90828109044453173665180803912, −6.20685448802671352741010106426, −5.33597418528288461723941794721, −4.691952841556892169461916419820, −3.23374863335030904512826208174, −2.548719480702293325461319323115, −1.70179285017308253165723015075, 1.22178951482956634581233357836, 1.960269475288315317391270816208, 3.21079910943586458730474455391, 4.29714598574544243906189922577, 4.95010229540183436185310285394, 5.89711050927036692898654430373, 6.633906242654624097825852144789, 7.840138406501360130902691298205, 8.60712599771980568519602243155, 9.98955726263620453604932730721, 10.69507659558237050593144549853, 11.30417702675562770412277771854, 12.61910266539710735777879489973, 13.15350030387633500920289160832, 13.72718018079743062402081929861, 14.54655560726093878292422082973, 15.52535386685011109370531282479, 16.26952610956025840236725033706, 17.16427487980949515353885409266, 17.79793099291924222926708792599, 19.048293589112369441205375375464, 19.914237705050388213244034651869, 20.75389216604196644601653015882, 21.24934324242288390279578018474, 21.67343731097511017136857924997

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