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Properties

Label 1-837-837.481-r0-0-0
Degree $1$
Conductor $837$
Sign $0.561 - 0.827i$
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  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (0.766 − 0.642i)5-s + (0.961 + 0.275i)7-s + (−0.104 − 0.994i)8-s + (−0.978 + 0.207i)10-s + (−0.241 − 0.970i)11-s + (−0.882 + 0.469i)13-s + (−0.719 − 0.694i)14-s + (−0.374 + 0.927i)16-s + (0.913 + 0.406i)17-s + (0.669 + 0.743i)19-s + (0.961 + 0.275i)20-s + (−0.241 + 0.970i)22-s + (0.961 − 0.275i)23-s + ⋯
L(s)  = 1  + (−0.882 − 0.469i)2-s + (0.559 + 0.829i)4-s + (0.766 − 0.642i)5-s + (0.961 + 0.275i)7-s + (−0.104 − 0.994i)8-s + (−0.978 + 0.207i)10-s + (−0.241 − 0.970i)11-s + (−0.882 + 0.469i)13-s + (−0.719 − 0.694i)14-s + (−0.374 + 0.927i)16-s + (0.913 + 0.406i)17-s + (0.669 + 0.743i)19-s + (0.961 + 0.275i)20-s + (−0.241 + 0.970i)22-s + (0.961 − 0.275i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117337144 - 0.5918193899i\)
\(L(\frac12)\) \(\approx\) \(1.117337144 - 0.5918193899i\)
\(L(1)\) \(\approx\) \(0.8992958307 - 0.2743756374i\)
\(L(1)\) \(\approx\) \(0.8992958307 - 0.2743756374i\)

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 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (-0.241 - 0.970i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.961 - 0.275i)T \)
29 \( 1 + (-0.882 - 0.469i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.990 + 0.139i)T \)
43 \( 1 + (0.848 - 0.529i)T \)
47 \( 1 + (-0.615 - 0.788i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.848 + 0.529i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (-0.882 - 0.469i)T \)
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

−22.43641427755847198902889163335, −21.19866451137826254730805933025, −20.64496740293803757406444680958, −19.80760898344810122055721629591, −18.81700214123840842636780515486, −18.09020634319230587727660415686, −17.46352139884300428760557689949, −17.06920351229836804690413281833, −15.84681268191518291928324019741, −14.84340960080538554927058151086, −14.58879562349212652167385361580, −13.62299748891919724253378551479, −12.40780017357340059269218153386, −11.260148815837301843039038200578, −10.68995499013404756833694306066, −9.70097083203979060500106773765, −9.33427966130235071735107538624, −7.836010366710770215138802385419, −7.43493515905160353060787010389, −6.62147794944079507626092147931, −5.28617392218690964044913572594, −4.98114867243025444868175113090, −3.02822713610854147097192512450, −2.112103857892394767891895933434, −1.14194537323711815845788035409, 0.9251472473505613325876531369, 1.814429657344833487614442240475, 2.7151978758746640178332074737, 3.96786954571415014883445778678, 5.21962755282944140975198765459, 5.92987037391845832680179491882, 7.31911495643245854764104907443, 8.07401868959901675305259457929, 8.87713066255706947456531115553, 9.562195669952932691123330828305, 10.46258377863014230678526114776, 11.29496978642069616852284880102, 12.14356757690944422057715049945, 12.817828152173202658432928492955, 13.927346487919564024642700989498, 14.6684392334344191379043391296, 15.90822790123515263388070872016, 16.81009852640682627692172657505, 17.11140627521526053750142803229, 18.103475353117953829929856380853, 18.76246180323952729043259389712, 19.515708643261696265289143618112, 20.61120046047480651366439627425, 21.15865547929851020393408726483, 21.514883399656705480739070329506

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