Properties

Label 1-837-837.349-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: $\chi_{837} (349, \cdot )$
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

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

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