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Properties

Label 1-837-837.290-r0-0-0
Degree $1$
Conductor $837$
Sign $0.181 + 0.983i$
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  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.374 − 0.927i)11-s + (−0.961 − 0.275i)13-s + (−0.990 + 0.139i)14-s + (0.559 + 0.829i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.615 − 0.788i)20-s + (−0.990 + 0.139i)22-s + (0.990 − 0.139i)23-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (−0.882 − 0.469i)4-s + (−0.173 + 0.984i)5-s + (−0.374 − 0.927i)7-s + (−0.669 + 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.374 − 0.927i)11-s + (−0.961 − 0.275i)13-s + (−0.990 + 0.139i)14-s + (0.559 + 0.829i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (0.615 − 0.788i)20-s + (−0.990 + 0.139i)22-s + (0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (290, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1787348468 + 0.1487098168i\)
\(L(\frac12)\) \(\approx\) \(0.1787348468 + 0.1487098168i\)
\(L(1)\) \(\approx\) \(0.6804731767 - 0.3106859780i\)
\(L(1)\) \(\approx\) \(0.6804731767 - 0.3106859780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 + (-0.374 - 0.927i)T \)
11 \( 1 + (-0.374 - 0.927i)T \)
13 \( 1 + (-0.961 - 0.275i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.990 - 0.139i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.559 + 0.829i)T \)
43 \( 1 + (0.719 + 0.694i)T \)
47 \( 1 + (-0.438 + 0.898i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (0.241 + 0.970i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.0348 + 0.999i)T \)
83 \( 1 + (-0.719 - 0.694i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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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

−22.27114418581190267512355830762, −21.08217624820813644856608384646, −20.71658708808897080345989445869, −19.34475113343539265168581990660, −18.73945837154701940301804209535, −17.72707613319804943595662929020, −17.00022987418383868591353003580, −16.303237099361914594436445285942, −15.49095667633959393159700718997, −14.97967788328898308293177353076, −13.93384686886645705929511004075, −13.0109687083417732752228135268, −12.261755777967972160058562098882, −11.91726792043515546955938932521, −10.01568280776849600024420199225, −9.379227705458800964567149600969, −8.675578037375206478709209697006, −7.68848821240691490829738080443, −7.01733554549047541286861055184, −5.76561799718499988113867545588, −5.131340721855797380387963113036, −4.44286144702105178286905862773, −3.211247024378378803302052098451, −1.93628319467738991940538699285, −0.09967539429190812140300733156, 1.29912843918233579136075035532, 2.80368174876354367137681794871, 3.182860202025205165056576696715, 4.221749370270646616204240413242, 5.247178509938208327234309945057, 6.35825852620610494622438943287, 7.27467049246044877907898087120, 8.28542276635956487582806097325, 9.413289225139513879889157974699, 10.33268209631269274303308927811, 10.80595402645652939579360436354, 11.46721394158328598080044394531, 12.695999380860477194970616043011, 13.234199941280800350469914733311, 14.21902928372182039903450410792, 14.72244826542823301228038796852, 15.73761841561003587621524666430, 16.977197835249805679152404907868, 17.62114399132685624505999689459, 18.61754809108834049810397228713, 19.39773355729714044582642214482, 19.65440337116646694070208882348, 20.77125249677341265111705695237, 21.63226670803103030275001732057, 22.18511692229752866062442619287

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