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Properties

Label 1-837-837.284-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.336 - 0.941i$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + 10-s + (−0.766 − 0.642i)11-s + (0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.939 − 0.342i)20-s + (−0.939 − 0.342i)22-s + (−0.766 + 0.642i)23-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.939 + 0.342i)5-s + (0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + 10-s + (−0.766 − 0.642i)11-s + (0.766 − 0.642i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)16-s + (0.5 − 0.866i)17-s + 19-s + (0.939 − 0.342i)20-s + (−0.939 − 0.342i)22-s + (−0.766 + 0.642i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.336 - 0.941i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.336 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.957204820 - 4.195946081i\)
\(L(\frac12)\) \(\approx\) \(2.957204820 - 4.195946081i\)
\(L(1)\) \(\approx\) \(2.072806847 - 1.078619966i\)
\(L(1)\) \(\approx\) \(2.072806847 - 1.078619966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.173 - 0.984i)T \)
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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

−21.91299629315636510173472826732, −21.63336824568291613160262561324, −20.76756901951871662573356724595, −20.26856496279711588442961900909, −18.8740544102520665420566267963, −18.02812667952347935373596638524, −17.37687521436162298761004481712, −16.23878799774555385759352159510, −15.822759002611398549519749655391, −14.78214188928790264577613703278, −14.09653651058447820776199195537, −13.32778355811064422469166762941, −12.45907514170051094149453297037, −11.98947649736600801254817799680, −10.78615984869558886219626878297, −9.90266397168073658381411346736, −8.75231830755228232709781590026, −8.08123137681288995707318866, −6.84175861189477752662113121898, −5.987905316699143013805863476120, −5.36275581729938509091991190027, −4.58280989458584700443798730644, −3.33321396963263875567211886915, −2.26900623344105992688266055583, −1.58359544751612106711151175314, 0.7508194055554759428287027791, 1.64415469665127897638884178286, 2.99624229392499190350697138967, 3.42780761974949609464326504400, 4.8574156052245665288065163580, 5.50474787659097812253858168804, 6.37281874743350171767479214308, 7.27848307970252198999729987828, 8.24551059187811500522042564724, 9.929443929854700899583998728081, 10.12848862969941064929432483465, 11.12307364065278881560714195195, 11.82728791939902899443450485868, 13.15520632876235110714198520701, 13.655107804244844025866228270987, 13.98206720775902805459612009339, 15.0869254682888946976492710995, 16.02862730649706089040844200155, 16.661588929007338715881234331666, 17.94337589274243932546476170304, 18.38783795329888885874524556635, 19.56448927450744417396422492188, 20.435289174404726254343902694514, 20.93000958134853434418850004861, 21.64188195823589652561137375836

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