Properties

Label 1-837-837.826-r0-0-0
Degree $1$
Conductor $837$
Sign $0.372 - 0.928i$
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  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.939 + 0.342i)5-s + (0.990 + 0.139i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (0.990 + 0.139i)11-s + (−0.241 − 0.970i)13-s + (−0.615 − 0.788i)14-s + (−0.997 + 0.0697i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.374 − 0.927i)20-s + (−0.615 − 0.788i)22-s + (−0.615 − 0.788i)23-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.0348 + 0.999i)4-s + (−0.939 + 0.342i)5-s + (0.990 + 0.139i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (0.990 + 0.139i)11-s + (−0.241 − 0.970i)13-s + (−0.615 − 0.788i)14-s + (−0.997 + 0.0697i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.374 − 0.927i)20-s + (−0.615 − 0.788i)22-s + (−0.615 − 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.372 - 0.928i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (826, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.372 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7569045704 - 0.5119581913i\)
\(L(\frac12)\) \(\approx\) \(0.7569045704 - 0.5119581913i\)
\(L(1)\) \(\approx\) \(0.7218215929 - 0.2233909433i\)
\(L(1)\) \(\approx\) \(0.7218215929 - 0.2233909433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (0.990 + 0.139i)T \)
11 \( 1 + (0.990 + 0.139i)T \)
13 \( 1 + (-0.241 - 0.970i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.615 - 0.788i)T \)
29 \( 1 + (-0.719 - 0.694i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.997 - 0.0697i)T \)
43 \( 1 + (0.961 - 0.275i)T \)
47 \( 1 + (0.559 + 0.829i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (-0.719 + 0.694i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.848 + 0.529i)T \)
83 \( 1 + (0.961 - 0.275i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.374 - 0.927i)T \)
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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

−22.52311543068513987621786578332, −21.41941111503916270065445726078, −20.31148512518259829961990726721, −19.935887259469034026930219707107, −18.853501586205273019236059055694, −18.44530855249959944511360170174, −17.27284909400623357675723669145, −16.701593617208936494445315250747, −16.069938499428182387151401462582, −15.07983948597480129711023508983, −14.38372635513874956232576617956, −13.829546831069355187885428697522, −12.202063125367088888009117201217, −11.56046948195499693454533773672, −10.924284742404787369071207119021, −9.61210437428755370981514365713, −9.00586721986193918588870707765, −8.03156406471225822923129224053, −7.50477729387288738738469718753, −6.61272254900879447480579591718, −5.441924540394717616075360635494, −4.57667360287037738581570303357, −3.7316294074846057390676572686, −1.89944286114295580907726833669, −1.04796401892427877080031050649, 0.67084531424663758312888385429, 1.88008204848984215189344345427, 2.93146002632096638115168977608, 3.96847125855842513127418457289, 4.63041427735503667833750328347, 6.185505085029598824938520198490, 7.34415465719433247245656458244, 7.98599930305539829547965497758, 8.640001592838319880543256738532, 9.6620644824869173661957609576, 10.781263170467776928470953144259, 11.15014719212984425147245013785, 12.11509785598814346564332380103, 12.59217849759719188658025221863, 13.89638472413159406762489445278, 14.93318310745934120841781912948, 15.442211937562758254911103987887, 16.677669838893953174815373850, 17.33936923875359317481110900771, 18.06634332576165471598630913874, 18.88026893200097363767777049822, 19.66283598369678304457206411584, 20.19304279443400337952019530125, 20.98318291514531597755357239800, 22.11388388121691222223314211322

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