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Properties

Label 1-837-837.250-r0-0-0
Degree $1$
Conductor $837$
Sign $0.561 + 0.827i$
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  + (0.848 + 0.529i)2-s + (0.438 + 0.898i)4-s + (−0.939 + 0.342i)5-s + (−0.719 − 0.694i)7-s + (−0.104 + 0.994i)8-s + (−0.978 − 0.207i)10-s + (0.961 − 0.275i)11-s + (0.848 − 0.529i)13-s + (−0.241 − 0.970i)14-s + (−0.615 + 0.788i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.719 − 0.694i)20-s + (0.961 + 0.275i)22-s + (−0.719 + 0.694i)23-s + ⋯
L(s)  = 1  + (0.848 + 0.529i)2-s + (0.438 + 0.898i)4-s + (−0.939 + 0.342i)5-s + (−0.719 − 0.694i)7-s + (−0.104 + 0.994i)8-s + (−0.978 − 0.207i)10-s + (0.961 − 0.275i)11-s + (0.848 − 0.529i)13-s + (−0.241 − 0.970i)14-s + (−0.615 + 0.788i)16-s + (0.913 − 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.719 − 0.694i)20-s + (0.961 + 0.275i)22-s + (−0.719 + 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.561 + 0.827i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.561 + 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.855469667 + 0.9827856638i\)
\(L(\frac12)\) \(\approx\) \(1.855469667 + 0.9827856638i\)
\(L(1)\) \(\approx\) \(1.438030122 + 0.5129393025i\)
\(L(1)\) \(\approx\) \(1.438030122 + 0.5129393025i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.848 + 0.529i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.719 - 0.694i)T \)
11 \( 1 + (0.961 - 0.275i)T \)
13 \( 1 + (0.848 - 0.529i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.719 + 0.694i)T \)
29 \( 1 + (0.848 + 0.529i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.374 + 0.927i)T \)
43 \( 1 + (0.0348 - 0.999i)T \)
47 \( 1 + (0.990 + 0.139i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.0348 + 0.999i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.913 + 0.406i)T \)
79 \( 1 + (-0.997 + 0.0697i)T \)
83 \( 1 + (0.848 + 0.529i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.961 - 0.275i)T \)
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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

−22.086149130677445767162346117108, −21.25522365426198048987807314674, −20.416619143524917263287050871037, −19.752388234556071475753373578717, −18.94721266463309342976551962505, −18.55200885441627381287241782333, −16.97425064901924479428861338705, −15.952674761614961978881122245026, −15.74870021068058020854843078199, −14.54882051649911095048057364446, −14.03133631835289483052050004663, −12.795964950708251449100015406192, −12.1541884678506676511094273587, −11.81070126487732332095600233595, −10.73209567748826574536924058503, −9.72214933136494952844080825031, −8.93349784476694766905941541074, −7.85056908228549780455454235838, −6.610503690552563708774971746555, −6.00776037273532780117332574177, −4.92157643963177574271824341717, −3.79054942413455491903199959019, −3.51133742400703051817064534003, −2.10382208325477661129567694028, −0.96982528764837570016125540313, 1.036735741183501580936085640699, 3.01197039457341560995107091256, 3.48913594726769388754910064602, 4.23794947641234627632495285679, 5.406634231476341609357958871214, 6.45252869762960638177097186668, 7.06765074453611598089930741576, 7.87637892877905948157601422727, 8.768948365872395580955309390649, 10.03307351153058662197760364174, 11.091786217987192340114721619530, 11.80093951169512430611055760672, 12.50466657065187122366815572780, 13.64876830196101177885426048330, 14.013062295946707283841768437108, 15.086975474569446428539733022513, 15.85623322634993588156980165012, 16.32562375854283405210588719653, 17.20777921108630538123392477559, 18.20026611053894137992197446923, 19.28530812669379481656801032901, 20.025508440883585816522406641288, 20.57081570146504593389402594777, 21.84448227867518323389300156340, 22.399723363613075969885077111098

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