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Properties

Label 1-837-837.121-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.162 + 0.986i$
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  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.766 − 0.642i)5-s + (0.961 + 0.275i)7-s + (−0.104 − 0.994i)8-s + (0.669 + 0.743i)10-s + (0.961 + 0.275i)11-s + (−0.882 + 0.469i)13-s + (−0.241 + 0.970i)14-s + (0.990 − 0.139i)16-s + (−0.809 + 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.719 + 0.694i)20-s + (−0.241 + 0.970i)22-s + (−0.241 + 0.970i)23-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (−0.997 + 0.0697i)4-s + (0.766 − 0.642i)5-s + (0.961 + 0.275i)7-s + (−0.104 − 0.994i)8-s + (0.669 + 0.743i)10-s + (0.961 + 0.275i)11-s + (−0.882 + 0.469i)13-s + (−0.241 + 0.970i)14-s + (0.990 − 0.139i)16-s + (−0.809 + 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.719 + 0.694i)20-s + (−0.241 + 0.970i)22-s + (−0.241 + 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.162 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.162 + 0.986i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.162 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.052681771 + 1.240714278i\)
\(L(\frac12)\) \(\approx\) \(1.052681771 + 1.240714278i\)
\(L(1)\) \(\approx\) \(1.044102493 + 0.6113492401i\)
\(L(1)\) \(\approx\) \(1.044102493 + 0.6113492401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.0348 + 0.999i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (0.961 + 0.275i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.241 + 0.970i)T \)
29 \( 1 + (0.0348 + 0.999i)T \)
37 \( 1 + T \)
41 \( 1 + (0.990 + 0.139i)T \)
43 \( 1 + (0.848 - 0.529i)T \)
47 \( 1 + (-0.374 + 0.927i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (0.848 - 0.529i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.719 + 0.694i)T \)
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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

−21.804591578804551561121079822867, −21.21369711884848270243785084954, −20.3528112692367024009851394166, −19.621381103927959656422671067708, −18.80983816997122598827747675998, −17.84208088880148623311258068217, −17.51223410656626833054102884744, −16.687205169120808896039458205990, −14.92771712260636231160638807993, −14.57896007957626810393516461574, −13.76827443656085476747872058096, −13.01422531825473528394764391634, −11.9536867573515698557026877533, −11.19346598108550160501490180666, −10.57247796471414516713392885379, −9.71342997298418473521556607093, −8.91589839906474249703906393596, −7.95368574478785631864234760169, −6.78957354049137921758677244608, −5.773990440222179342157209613916, −4.69360215154020715615694473884, −4.003992603368240428284631684314, −2.568255844240327941926493008862, −2.14579824648605281196900892742, −0.813036955021255117967602421456, 1.28687275962433621824262752739, 2.21825345135983015764359463897, 4.110759548169935036560557537621, 4.62559979850596478717305993169, 5.5860880560801552897034396367, 6.352930164370832550270134872259, 7.30689817600751450362641397425, 8.308978135101205082427281575524, 9.05111060390891325544599143983, 9.6005591700611377818950312935, 10.801489756815660541853329822309, 12.030272970936356484780911860566, 12.737366931469090516565845116810, 13.64783327639712021366622750299, 14.56772512126486057849947163953, 14.83005609660597923310924864348, 16.0430016097197375153711275733, 16.88736123374379388637640364344, 17.52191804443844800879525649007, 17.8435359571550524605615375858, 19.10212189211040325787962947769, 19.885481694526973484705413424825, 21.02846461108835394925755820657, 21.815390015484984617338992533906, 22.13161183949356167879785093931

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