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Properties

Label 1-837-837.643-r1-0-0
Degree $1$
Conductor $837$
Sign $0.746 - 0.664i$
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  + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.766 − 0.642i)5-s + (−0.615 − 0.788i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (0.374 − 0.927i)11-s + (0.719 + 0.694i)13-s + (0.990 + 0.139i)14-s + (−0.997 − 0.0697i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.615 − 0.788i)20-s + (0.374 + 0.927i)22-s + (0.615 − 0.788i)23-s + ⋯
L(s)  = 1  + (−0.719 + 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.766 − 0.642i)5-s + (−0.615 − 0.788i)7-s + (0.669 + 0.743i)8-s + (−0.104 + 0.994i)10-s + (0.374 − 0.927i)11-s + (0.719 + 0.694i)13-s + (0.990 + 0.139i)14-s + (−0.997 − 0.0697i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (−0.615 − 0.788i)20-s + (0.374 + 0.927i)22-s + (0.615 − 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.746 - 0.664i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.746 - 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.889745275 - 0.7193154787i\)
\(L(\frac12)\) \(\approx\) \(1.889745275 - 0.7193154787i\)
\(L(1)\) \(\approx\) \(1.012516370 - 0.06299526348i\)
\(L(1)\) \(\approx\) \(1.012516370 - 0.06299526348i\)

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 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
11 \( 1 + (0.374 - 0.927i)T \)
13 \( 1 + (0.719 + 0.694i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.615 - 0.788i)T \)
29 \( 1 + (0.719 - 0.694i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.438 - 0.898i)T \)
43 \( 1 + (0.241 + 0.970i)T \)
47 \( 1 + (0.559 - 0.829i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.241 + 0.970i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (0.882 + 0.469i)T \)
83 \( 1 + (0.719 - 0.694i)T \)
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

−21.94937207716902252218250401005, −21.1550582397897896365076614014, −20.476408630455635676723643170968, −19.49843258315771202798910418920, −18.82225767186291211806996938310, −18.00934293928261486129296043516, −17.65526841156253705929748614023, −16.56210896584017940135243548288, −15.738919743402128492122274859949, −14.80236549276802485395128182049, −13.79371097173350216630274141220, −12.82162990137395620268381257199, −12.2794109841151406854423367764, −11.27362758640562251461598345302, −10.38790527902810810986706585006, −9.65195639092346260312386113547, −9.18345711626802044020072604813, −7.986000059351669191116449248662, −7.10982246450938697270132275429, −6.15109904413096303538846296607, −5.21175781972986911085246599505, −3.57651476607851818683264990707, −3.01659500021564981739448860229, −1.97707521986575090373908665342, −1.02590949842614315796174493382, 0.80616503448499765015018269707, 1.11419371198256498554991853560, 2.71177954034190104624947715635, 4.07227346132762825560257582657, 5.15800889142964305897476714559, 6.094683855646526460293675150354, 6.65381425944504119072671510591, 7.731851163483488863880028872192, 8.70929464115104873908832874230, 9.34483751428245480756540963608, 10.08642390073455953583102804684, 10.926300063595147973837957709455, 11.96915904191833357991133884619, 13.331675557550497815949169416671, 13.78823873326460466174600216070, 14.45825643706915807335847511176, 15.840852846265573467840221641698, 16.42543137253220395909271024216, 16.857804411009991963190873916799, 17.68014517174414622210543639600, 18.6737190526130601390496733149, 19.231262847520192935374144779212, 20.23071683009564159976546658955, 20.837419788421105211467531996060, 21.852518365348199540279620808687

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