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Properties

Label 1-837-837.767-r0-0-0
Degree $1$
Conductor $837$
Sign $0.949 - 0.314i$
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  + (−0.438 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.949 - 0.314i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9016441275 - 0.1455861287i\)
\(L(\frac12)\) \(\approx\) \(0.9016441275 - 0.1455861287i\)
\(L(1)\) \(\approx\) \(0.7649637654 + 0.08037404473i\)
\(L(1)\) \(\approx\) \(0.7649637654 + 0.08037404473i\)

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

−21.99768354757688547028490624088, −21.550765421989119619870599707608, −20.4750719451385717388443565989, −19.565276515565816812860753959868, −19.05331012611027609819150442329, −18.47695640118111432472859711118, −17.67544380414076720827675787670, −16.541503936035824798437223949456, −15.98769988595825755776012092942, −14.65348363544016275780188701173, −14.25967917058612347587758744225, −12.987464931870528651035771944242, −11.98263595546913082589571813037, −11.65858245178432917829049134090, −10.92199923925779633148094793693, −9.80048634065508225549402632986, −8.99740907550946142235753935667, −8.35990827063679552189822578826, −7.25649937870170038734523402223, −6.44090736562111926075731072234, −4.9545833702924774310365406949, −4.11177328224109989306278548026, −2.99159507940589791651640512596, −2.45968465993322004044040492160, −0.99717948114506112182218448441, 0.65372463392287463282506156923, 1.59286572552813147250364857289, 3.69849091191351541358355779493, 4.225474517363018958749989978804, 5.28949962543467609468789751886, 6.20488137736554156664914630111, 7.480155873470913368698792086288, 7.681081322050493582366821146389, 8.65418364357212400739417806220, 9.71870164144129364977916051178, 10.2954026226893635200994670804, 11.418000322215738748028937685634, 12.494071317975676005980911122248, 13.20417920078906581572112359992, 14.378708749867153372909462340759, 14.83076860876515630450811559005, 15.84712921717140638045913569390, 16.49295219691261387199884794741, 17.34870825608699848201352749361, 17.62091807135404860081419499362, 19.148662694354842551370465362663, 19.51892674959426152187657230857, 20.24506064587982580196934948072, 21.16921706391962108587096431419, 22.66872006136162187922834930437

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