Properties

Label 1-837-837.500-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.999 + 0.0204i$
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  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (0.882 + 0.469i)11-s + (0.559 + 0.829i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.848 + 0.529i)20-s + (−0.882 + 0.469i)22-s + (−0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (−0.559 + 0.829i)2-s + (−0.374 − 0.927i)4-s + (−0.173 − 0.984i)5-s + (0.848 − 0.529i)7-s + (0.978 + 0.207i)8-s + (0.913 + 0.406i)10-s + (0.882 + 0.469i)11-s + (0.559 + 0.829i)13-s + (−0.0348 + 0.999i)14-s + (−0.719 + 0.694i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (−0.848 + 0.529i)20-s + (−0.882 + 0.469i)22-s + (−0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.999 + 0.0204i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (500, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.999 + 0.0204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001729261284 + 0.1691788983i\)
\(L(\frac12)\) \(\approx\) \(0.001729261284 + 0.1691788983i\)
\(L(1)\) \(\approx\) \(0.7261338365 + 0.1256324574i\)
\(L(1)\) \(\approx\) \(0.7261338365 + 0.1256324574i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.559 + 0.829i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
11 \( 1 + (0.882 + 0.469i)T \)
13 \( 1 + (0.559 + 0.829i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.848 - 0.529i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (0.438 + 0.898i)T \)
47 \( 1 + (0.241 + 0.970i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.438 + 0.898i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.669 + 0.743i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (-0.559 + 0.829i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−21.548393437885975168272997079012, −20.601280372829045275545392930966, −20.02455648839773795326644039153, −18.86895294208245804969283789759, −18.58582905472199331629460647761, −17.72808288111406654952859161914, −17.1083517400785505092974287333, −15.88573010486834402291995302118, −15.121084248196056753028543534416, −14.05997732824205204013921313343, −13.52627394072046808142135482158, −12.09388028790316828910764636723, −11.678671675318027835257274809, −10.910534396625405328358441598298, −10.183518825603617992788154468246, −9.17519500646043087931459035360, −8.282369820696623071585500398309, −7.65780194517528733218856826250, −6.51888073353391285789185978083, −5.45641249183617991059162278354, −4.07597152407149670458826453283, −3.391961431902426724505628958991, −2.333612832437717490892723935068, −1.45732408133685040512227994506, −0.04618035875134112849908509689, 1.26154065770314343623127438651, 1.818767900811923984970738225293, 4.210599362082527663560911612850, 4.38765365681141534665408597496, 5.54863678593285479906234248922, 6.599491885994710617862280804974, 7.34214270127408951999852881973, 8.43567962484188975154448258500, 8.83432685450557359709166217516, 9.710321687500118119690858354858, 10.86600588277708478982636226414, 11.5400382769073768985240375339, 12.72467266688295470437920818041, 13.65393396560516418544155878675, 14.38998138428170113074807847696, 15.173553198401548553149205776, 16.137425369326164208991009094407, 16.72570199033844485076769228901, 17.517218237635785987790482834373, 17.96453645147934359423645270674, 19.22566139950535649952702364357, 19.88502978733814855451426644741, 20.49067694926570654338425894616, 21.523811490047297018243048990983, 22.51536488725754472397849092771

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