Properties

Label 1-837-837.221-r1-0-0
Degree $1$
Conductor $837$
Sign $0.517 - 0.855i$
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  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (−0.766 + 0.642i)5-s + (0.0348 + 0.999i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.848 + 0.529i)11-s + (0.438 − 0.898i)13-s + (0.882 − 0.469i)14-s + (−0.241 − 0.970i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.0348 − 0.999i)20-s + (0.848 + 0.529i)22-s + (−0.0348 + 0.999i)23-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (−0.766 + 0.642i)5-s + (0.0348 + 0.999i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (−0.848 + 0.529i)11-s + (0.438 − 0.898i)13-s + (0.882 − 0.469i)14-s + (−0.241 − 0.970i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.0348 − 0.999i)20-s + (0.848 + 0.529i)22-s + (−0.0348 + 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.517 - 0.855i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.517 - 0.855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5483367518 - 0.3091531188i\)
\(L(\frac12)\) \(\approx\) \(0.5483367518 - 0.3091531188i\)
\(L(1)\) \(\approx\) \(0.5950937381 - 0.09160329805i\)
\(L(1)\) \(\approx\) \(0.5950937381 - 0.09160329805i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.438 - 0.898i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.0348 + 0.999i)T \)
11 \( 1 + (-0.848 + 0.529i)T \)
13 \( 1 + (0.438 - 0.898i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.0348 + 0.999i)T \)
29 \( 1 + (-0.438 - 0.898i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.719 + 0.694i)T \)
43 \( 1 + (-0.997 - 0.0697i)T \)
47 \( 1 + (-0.961 - 0.275i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.997 - 0.0697i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (-0.438 - 0.898i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (0.848 - 0.529i)T \)
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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

−22.47384935398199924605730226782, −21.03061732461952712845924261840, −20.4051085788421593584098024473, −19.5696485756801402872742150140, −18.7270797811178802574382602326, −18.13573662523280312144452941050, −16.91324784258275770032825412747, −16.4308460214356692033347400436, −15.96431212910401025948789233844, −14.96934805632973879430985366588, −13.98368937759645728656848427769, −13.43225839746969580211039364361, −12.43674858106921555921512138464, −11.17093357248697163124625362532, −10.58419494015460667163186049142, −9.48108868046544912639423500229, −8.58899227139822276843401521677, −7.96716213311168079911674455713, −7.13681625929680732434662334963, −6.32170888615010213398423416166, −5.07309361271005392905903643187, −4.44233224121140564337750556412, −3.48756596123391296886523812673, −1.63175072743111341022909828886, −0.55563194848359747856645838564, 0.29935311318234214083259472291, 1.92377454209255130418945599052, 2.73032011540460650457401355188, 3.54110919377341982411616612676, 4.611570651113932377025878269634, 5.65319266358407448889980863144, 6.98013392022999048408788838051, 7.95375381028047746799697955397, 8.487026571861072839850564315104, 9.55340282472195099322506769511, 10.435213646240335894457197369188, 11.22099912167034748363943042207, 11.761389357040120108928706863226, 12.87194489962564192341639999390, 13.2298950857923752452021333648, 14.71082129972235927662995910048, 15.447976353984644214796274586351, 15.98657469496582659999504615422, 17.52908089588250324926903291963, 17.92220939784770563891935076209, 18.675662997113226569300704417453, 19.48287264522634754556960865414, 19.9933069527810982858261397226, 21.06528779313644859317973455601, 21.67985612776281302673469097334

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