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Properties

Label 1-837-837.292-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.221 - 0.975i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.961 − 0.275i)11-s + (−0.0348 + 0.999i)13-s + (−0.241 + 0.970i)14-s + (−0.374 + 0.927i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.719 + 0.694i)20-s + (0.719 + 0.694i)22-s + (−0.961 + 0.275i)23-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (0.173 + 0.984i)5-s + (−0.241 − 0.970i)7-s + (−0.104 − 0.994i)8-s + (0.309 − 0.951i)10-s + (−0.961 − 0.275i)11-s + (−0.0348 + 0.999i)13-s + (−0.241 + 0.970i)14-s + (−0.374 + 0.927i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (−0.719 + 0.694i)20-s + (0.719 + 0.694i)22-s + (−0.961 + 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.221 - 0.975i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (292, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.221 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3367042485 - 0.4217848862i\)
\(L(\frac12)\) \(\approx\) \(0.3367042485 - 0.4217848862i\)
\(L(1)\) \(\approx\) \(0.6128122097 - 0.05120861309i\)
\(L(1)\) \(\approx\) \(0.6128122097 - 0.05120861309i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.882 - 0.469i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
11 \( 1 + (-0.961 - 0.275i)T \)
13 \( 1 + (-0.0348 + 0.999i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.961 + 0.275i)T \)
29 \( 1 + (0.882 + 0.469i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.615 + 0.788i)T \)
43 \( 1 + (0.882 + 0.469i)T \)
47 \( 1 + (-0.615 - 0.788i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−22.27815293104085310460013966033, −20.912622508910602501891179881500, −20.63159757021019147878820504836, −19.72630428985613064660724623973, −18.782171407761030261664909256814, −18.0688077224926066089140378768, −17.54705046695869597854255355874, −16.37074679884090431668550503706, −15.92677510318823195442075399003, −15.30714452992145628167140142153, −14.234834553585628453940182252999, −13.19790822156206667185297723300, −12.27378991537495474925093572561, −11.641378382507572967161437934847, −10.15078348577435319504943889457, −9.89987163750014512981703970010, −8.76910857918976809720971347918, −8.19051252631992243384518125161, −7.43897162883846844030765460401, −6.029373937501429600853440642849, −5.533346598438518613006462315901, −4.710127975256407387017077278975, −2.91388596144291094944566584624, −2.03998753740595780370333055721, −0.79492303631551003083375428284, 0.20606217012958225496799787665, 1.559168366506637677197240095247, 2.59120962238969730855134917020, 3.44504782176767224368154305255, 4.3807033465765224920841711446, 6.04749747893810914701422616550, 6.87127385034395294739953964207, 7.54286947553178183298514202887, 8.412903575614274883350922823115, 9.598878127864934047935589948541, 10.2231984183268190151496427556, 10.91291262463354628983298403283, 11.52903457181570111574635349000, 12.72881794686356096635213500878, 13.5540713696631759821729457294, 14.33379537230951324661310826329, 15.52963814349501080289241266631, 16.222301540401926770547166940838, 17.08159713807840282970545671600, 17.90074638979399196267910166173, 18.4433247458723012094771181149, 19.48223998888470501662465200751, 19.74005337046690590236638476395, 20.94751141196018841892368462792, 21.60209050274834390458168788410

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