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Properties

Label 1-837-837.22-r1-0-0
Degree $1$
Conductor $837$
Sign $0.909 + 0.414i$
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  + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.173 + 0.984i)5-s + (−0.374 + 0.927i)7-s + (0.669 + 0.743i)8-s + (−0.809 − 0.587i)10-s + (0.615 + 0.788i)11-s + (−0.961 + 0.275i)13-s + (−0.374 − 0.927i)14-s + (−0.997 − 0.0697i)16-s + (−0.669 − 0.743i)17-s + (−0.809 − 0.587i)19-s + (0.990 − 0.139i)20-s + (−0.990 − 0.139i)22-s + (0.615 − 0.788i)23-s + ⋯
L(s)  = 1  + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.173 + 0.984i)5-s + (−0.374 + 0.927i)7-s + (0.669 + 0.743i)8-s + (−0.809 − 0.587i)10-s + (0.615 + 0.788i)11-s + (−0.961 + 0.275i)13-s + (−0.374 − 0.927i)14-s + (−0.997 − 0.0697i)16-s + (−0.669 − 0.743i)17-s + (−0.809 − 0.587i)19-s + (0.990 − 0.139i)20-s + (−0.990 − 0.139i)22-s + (0.615 − 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.909 + 0.414i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.909 + 0.414i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9127852917 + 0.1982906467i\)
\(L(\frac12)\) \(\approx\) \(0.9127852917 + 0.1982906467i\)
\(L(1)\) \(\approx\) \(0.6256428429 + 0.3105027594i\)
\(L(1)\) \(\approx\) \(0.6256428429 + 0.3105027594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.719 + 0.694i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.374 + 0.927i)T \)
11 \( 1 + (0.615 + 0.788i)T \)
13 \( 1 + (-0.961 + 0.275i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.615 - 0.788i)T \)
29 \( 1 + (0.719 - 0.694i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.559 + 0.829i)T \)
43 \( 1 + (0.719 - 0.694i)T \)
47 \( 1 + (0.559 - 0.829i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.0348 - 0.999i)T \)
83 \( 1 + (0.241 + 0.970i)T \)
89 \( 1 + (0.978 + 0.207i)T \)
97 \( 1 + (-0.374 + 0.927i)T \)
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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

−21.71140502915828043238143058791, −20.93836313438711709538502091414, −20.155135862683422424761795684919, −19.475613135167017109921574974363, −19.11458121430736373496964990585, −17.52757943324577319072981289192, −17.290537036137534516253120966389, −16.55959070219647075960318657739, −15.84587876809435228961838313693, −14.47288155496266490175295136207, −13.441511019002290372852696555505, −12.84253717187710226360414462381, −12.130311225207121481877280265288, −11.097013196612667783972985129463, −10.34990115284037494550660741965, −9.50705033006752433865991079080, −8.77382120802555408198654783499, −7.98539207185494655942008326495, −7.0313951278665429477053290477, −5.97339795744297341363479616596, −4.55376183563575418378960007764, −3.916162552892047512737136810646, −2.798638136107351646408791916843, −1.50171274194633610047392936945, −0.76193710323805916570802821997, 0.36356604722597171881635392760, 2.18701209230255622777460189518, 2.53736629593222888778502359650, 4.31928467637866487930908991016, 5.26561867922501254151070627026, 6.50831247615155649769782016502, 6.73081961875051226714430633332, 7.71245141606199084972447534428, 8.94024318490668150162396585138, 9.4546097963070594105874766205, 10.26679412717502653319740349452, 11.21593527946283843782455647366, 12.0469822804074621652382383822, 13.19564367094769560603304482299, 14.365171370171728399724266130391, 14.84650148293476787594746242025, 15.47539609208785858519979038245, 16.366134826929658157745384823946, 17.481999905583254651359463754473, 17.7970551689939780962993577694, 18.83422738867358614967544658246, 19.28691336960042006166133275063, 20.06026541609208918583508319858, 21.32610201802014948611578141584, 22.28705492422046925913020920330

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