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Properties

Label 1-837-837.283-r0-0-0
Degree $1$
Conductor $837$
Sign $0.0204 - 0.999i$
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  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (0.173 + 0.984i)5-s + (0.848 − 0.529i)7-s + (−0.978 − 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.882 − 0.469i)11-s + (0.559 + 0.829i)13-s + (0.0348 − 0.999i)14-s + (−0.719 + 0.694i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.848 − 0.529i)20-s + (−0.882 + 0.469i)22-s + (0.848 + 0.529i)23-s + ⋯
L(s)  = 1  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (0.173 + 0.984i)5-s + (0.848 − 0.529i)7-s + (−0.978 − 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.882 − 0.469i)11-s + (0.559 + 0.829i)13-s + (0.0348 − 0.999i)14-s + (−0.719 + 0.694i)16-s + (0.669 − 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.848 − 0.529i)20-s + (−0.882 + 0.469i)22-s + (0.848 + 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0204 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0204 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.0204 - 0.999i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.0204 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.456087315 - 1.426621719i\)
\(L(\frac12)\) \(\approx\) \(1.456087315 - 1.426621719i\)
\(L(1)\) \(\approx\) \(1.306032432 - 0.6890169268i\)
\(L(1)\) \(\approx\) \(1.306032432 - 0.6890169268i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.559 - 0.829i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
11 \( 1 + (-0.882 - 0.469i)T \)
13 \( 1 + (0.559 + 0.829i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.848 + 0.529i)T \)
29 \( 1 + (0.559 - 0.829i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.961 - 0.275i)T \)
43 \( 1 + (0.438 + 0.898i)T \)
47 \( 1 + (-0.241 - 0.970i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.438 - 0.898i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (0.559 - 0.829i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−22.51271390423275096766033572068, −21.36782729014562714669811950243, −20.9264954207331494405568052980, −20.42971572403127095285507773559, −18.94256546168274576930572131528, −18.01613488392945886725266296059, −17.4864230748612414668173260456, −16.60246535852353739392020553119, −15.8648934744522177130512880084, −15.09423838775151343619032612899, −14.41285493073989319256007739794, −13.39870835193273059791607205136, −12.55742232351733115881898937944, −12.28515135173012843124036975402, −10.94811397419544197125312916745, −9.885241911256790580474033570, −8.584964255619929672241627445288, −8.30633041969224526139374785797, −7.475121239427224004356631648012, −6.10291530533792303405467010383, −5.372730502835351736726482461596, −4.85724999226789471344988313639, −3.78796572408798777759028926909, −2.57451351637564277594780370750, −1.257871732863056596855286329965, 0.8795231133620796850334149851, 2.137625924192695438418261292623, 2.930200298126328828002856111263, 3.89010595766019615149556700167, 4.881524782872774957953244308806, 5.725753004200118020912538471243, 6.796276599821135291897338784662, 7.6743191681589199420487607265, 8.91193769316972011745936577184, 9.851407595861589551610574669326, 10.83437003595072914736908925910, 11.13313555027048756160046759234, 11.92749426767597916969767445123, 13.284113173748909924717055672945, 13.7612842976065356848481913257, 14.410500977680309915993083292657, 15.25160192290408265426324933198, 16.129291873991461863216226623469, 17.5105852354226040963866829457, 18.10636076942974934079317117861, 18.90471834114979405308787079063, 19.4697814731229067597596776069, 20.640370635601307714688201209422, 21.29731344080284593878338464456, 21.590883440630848901028392356106

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