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Properties

Label 1-837-837.286-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.964 + 0.262i$
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  + (−0.719 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.964 + 0.262i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.964 + 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07040439579 - 0.5274435961i\)
\(L(\frac12)\) \(\approx\) \(-0.07040439579 - 0.5274435961i\)
\(L(1)\) \(\approx\) \(0.5200199737 - 0.3783747473i\)
\(L(1)\) \(\approx\) \(0.5200199737 - 0.3783747473i\)

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

−22.71123785602371230228422864796, −21.758196274281407782212263906039, −21.16951814556393008137282522031, −19.78791011383049031002830026883, −19.11717092078373767399473272540, −18.399538841044106883886957122854, −18.07877332999676443019146917823, −16.94737762382416945697361332205, −16.10776212892111413666194906159, −15.304568082813574964576746560939, −14.8986735105716246223321012713, −13.787494385095804078380436415032, −13.10227158915355309903526993244, −11.69849122443135839327339289480, −10.84116496945531202647567377790, −10.27176708480830223856611848956, −9.26131793858568229543692045631, −8.42882655730484669942435484096, −7.72271244967823548966560584184, −6.53614559879888272021014590062, −5.99848296309215583810186353738, −5.314160499481087991379600809596, −3.62165369859846056576999626841, −2.66282099358766411391537366688, −1.53726332885228548516869891627, 0.31103204005538254314041095442, 1.43938693321161648592068679197, 2.38658657872504384077903990995, 3.82583773378943822866706911555, 4.30425104794596087086149939025, 5.617095245057302919439991096296, 6.885913207577910873256804485048, 7.76671638007900117003424362491, 8.51911951598281724930751015706, 9.456167591113796861380203561145, 10.14068190025303725123589629282, 10.84219437149004148296177614862, 11.98192980704962635310095710349, 12.66661595083843361511620441601, 13.322588582370013909455165208788, 14.13308636860389269485147096285, 15.68293179695992636578933619412, 16.35944948903641669166160042798, 16.92097739520875175858950051259, 17.73815071787486883168638843198, 18.56407712229170547810581414001, 19.37518495007715999326299830020, 20.314709640823212965676441507570, 20.7600635543896422180680196419, 21.17315831118861705217567489348

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