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Properties

Label 1-837-837.380-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: $\chi_{837} (380, \cdot )$
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

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

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