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Properties

Label 1-837-837.158-r0-0-0
Degree $1$
Conductor $837$
Sign $-0.892 + 0.451i$
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  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 − 0.984i)5-s + (−0.241 − 0.970i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (−0.241 − 0.970i)11-s + (−0.0348 + 0.999i)13-s + (0.719 + 0.694i)14-s + (−0.615 − 0.788i)16-s + (−0.809 + 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.961 − 0.275i)20-s + (0.719 + 0.694i)22-s + (−0.719 − 0.694i)23-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.173 − 0.984i)5-s + (−0.241 − 0.970i)7-s + (0.104 + 0.994i)8-s + (0.669 + 0.743i)10-s + (−0.241 − 0.970i)11-s + (−0.0348 + 0.999i)13-s + (0.719 + 0.694i)14-s + (−0.615 − 0.788i)16-s + (−0.809 + 0.587i)17-s + (−0.978 + 0.207i)19-s + (−0.961 − 0.275i)20-s + (0.719 + 0.694i)22-s + (−0.719 − 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.892 + 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.892 + 0.451i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ -0.892 + 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01320357563 - 0.05534814102i\)
\(L(\frac12)\) \(\approx\) \(0.01320357563 - 0.05534814102i\)
\(L(1)\) \(\approx\) \(0.5145006917 - 0.06716390970i\)
\(L(1)\) \(\approx\) \(0.5145006917 - 0.06716390970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
11 \( 1 + (-0.241 - 0.970i)T \)
13 \( 1 + (-0.0348 + 0.999i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.719 - 0.694i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
37 \( 1 - T \)
41 \( 1 + (0.615 - 0.788i)T \)
43 \( 1 + (0.882 + 0.469i)T \)
47 \( 1 + (-0.990 + 0.139i)T \)
53 \( 1 + (0.913 + 0.406i)T \)
59 \( 1 + (-0.848 - 0.529i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.104 + 0.994i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (-0.882 - 0.469i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.961 + 0.275i)T \)
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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

−22.464297348667512822180238502141, −21.737643669223398667248692964422, −20.99866163554544100341904313977, −19.873741623184091585287189830802, −19.507134702211754560703948071028, −18.480777452541181858929138283929, −17.91478360784701619894076851349, −17.497481880432273764120220114181, −16.03262182766665340192127578176, −15.47661936913240241315583589659, −14.85383999522331194240028695076, −13.495236608109418990234858293647, −12.5513688998137817450194684347, −11.94280534667594344536964107242, −10.98801986887121132660550744401, −10.30169652161681537759587105751, −9.542290328820200836839582699822, −8.60859666566960486560871203136, −7.72413794615767060388004658939, −6.90906177217075606634950044923, −6.06320315030348964145511622932, −4.676655272992044921674490299459, −3.40425152971745331065741739642, −2.603133739552966828319757913447, −1.92699600627733946459874146097, 0.03556352940731866620592175108, 1.166009831083655647227985579041, 2.25907636675544369895028009909, 3.973901681442472920288294385270, 4.65815609417460631185709780873, 5.96417143598963849084489205330, 6.58424695128921777209031321130, 7.66852003288424891237097630744, 8.5071279243167779204508198173, 8.99105643449363683699987285111, 10.119314531333021171134545283, 10.794055553493702552061231557474, 11.69529116937705029199867480593, 12.80729488622099737184391020739, 13.74757742842049151914353152243, 14.37358694428547353867614777352, 15.69267830537105717309653942981, 16.15354607094195185202468463157, 16.95399552462713206527867519121, 17.33588104127579180407045598984, 18.50631877763605254935628554997, 19.54760173822077480400895827910, 19.62564654927164749445663366095, 20.82991564564129741769009528563, 21.35118933578023495473917306961

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