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Properties

Label 1-837-837.139-r1-0-0
Degree $1$
Conductor $837$
Sign $0.999 + 0.0236i$
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  + (0.848 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.999 + 0.0236i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ 0.999 + 0.0236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.309332277 + 0.01546762431i\)
\(L(\frac12)\) \(\approx\) \(1.309332277 + 0.01546762431i\)
\(L(1)\) \(\approx\) \(1.017279345 - 0.3908798892i\)
\(L(1)\) \(\approx\) \(1.017279345 - 0.3908798892i\)

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

−22.20544237471845664558011522498, −21.35540197982236933962245152755, −20.23123479236148865855759421432, −19.84361246902790405199015355952, −18.83281494646908119754805796654, −17.79856134196932481265428788487, −16.822582575653480925179614994284, −16.19774194507509618417429877019, −15.3827539689346867407360586449, −14.8695788733541505531637123365, −13.82173888110508565003332052238, −13.02963246386490898085643977702, −12.43627932265735854238679043858, −11.35251789495608155629919961192, −10.76885146924801680844599586252, −9.54295052739354101568711698585, −8.38066284043151625107378567497, −7.25016887444274027729583078887, −7.17275465465655492261556405868, −6.02087450280541743423686405556, −4.719790006182446158665170667195, −4.24496248677106017569668555970, −3.130054216608053925307643592554, −2.413293863224389365009414549836, −0.30978619342997385173927174850, 0.67986437330536669137621352417, 2.24361234255850512302991761503, 3.09020173602799010876793471468, 3.841697108828647709028387771871, 5.16455802290890250852729826394, 5.43435348537937175067120714831, 6.82886301421445044866866449201, 7.61336653591161041615061793201, 8.80996242374900908824471309518, 9.709829575540297999343925830323, 10.64128210865352904762824523883, 11.54225125768746056598869009591, 12.22689411472559769040683415114, 12.90709786141306589173804952305, 13.56171793693502047532739588045, 14.81030654045080154642221696448, 15.5140025979435233295298722031, 15.88069076493794512727678253414, 16.91262033841250635139749003774, 18.41959070667970415628493754586, 18.84830475113745976173747909298, 19.78228629476883699636737573955, 20.29957735338160065939512531355, 21.12664309460476310643315661275, 22.19532238138793886709836162043

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