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Properties

Label 1-837-837.50-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.201 - 0.979i$
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  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (−0.173 − 0.984i)5-s + (−0.997 − 0.0697i)7-s + (−0.913 + 0.406i)8-s + (−0.978 − 0.207i)10-s + (0.997 + 0.0697i)11-s + (−0.615 + 0.788i)13-s + (−0.438 + 0.898i)14-s + (0.0348 + 0.999i)16-s + (0.809 + 0.587i)17-s + (0.669 − 0.743i)19-s + (−0.559 + 0.829i)20-s + (0.438 − 0.898i)22-s + (−0.438 + 0.898i)23-s + ⋯
L(s)  = 1  + (0.374 − 0.927i)2-s + (−0.719 − 0.694i)4-s + (−0.173 − 0.984i)5-s + (−0.997 − 0.0697i)7-s + (−0.913 + 0.406i)8-s + (−0.978 − 0.207i)10-s + (0.997 + 0.0697i)11-s + (−0.615 + 0.788i)13-s + (−0.438 + 0.898i)14-s + (0.0348 + 0.999i)16-s + (0.809 + 0.587i)17-s + (0.669 − 0.743i)19-s + (−0.559 + 0.829i)20-s + (0.438 − 0.898i)22-s + (−0.438 + 0.898i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.201 - 0.979i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.201 - 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.182835312 - 1.450677237i\)
\(L(\frac12)\) \(\approx\) \(1.182835312 - 1.450677237i\)
\(L(1)\) \(\approx\) \(0.8799753827 - 0.6480885051i\)
\(L(1)\) \(\approx\) \(0.8799753827 - 0.6480885051i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.374 - 0.927i)T \)
5 \( 1 + (-0.173 - 0.984i)T \)
7 \( 1 + (-0.997 - 0.0697i)T \)
11 \( 1 + (0.997 + 0.0697i)T \)
13 \( 1 + (-0.615 + 0.788i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (-0.438 + 0.898i)T \)
29 \( 1 + (0.374 - 0.927i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.0348 + 0.999i)T \)
43 \( 1 + (0.990 - 0.139i)T \)
47 \( 1 + (0.882 + 0.469i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (0.374 + 0.927i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.961 + 0.275i)T \)
83 \( 1 + (-0.990 + 0.139i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (0.559 - 0.829i)T \)
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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

−22.28357639919204998115906070847, −21.97700367635855721960559370491, −20.618314812450337275406723961595, −19.648551427039850337536104457950, −18.77878550480360957504858103193, −18.17139189375351228329868112898, −17.21766899820801354561892751821, −16.3608759419755867872616339263, −15.80576778290444614457953085169, −14.71711292299931354559315398996, −14.37755224527728772499865582962, −13.49198466208602189911897913300, −12.317155097485847874094874495680, −11.99250021350986237318038187760, −10.471730568746661144474583300810, −9.751609749914602861831545407, −8.86937099026052160800416910510, −7.63391283210045766508461244208, −7.13946499573078970108157043232, −6.19912599538886193591195435927, −5.60718508040843142890895148799, −4.23557101264214152794179888456, −3.36342900049630469716067112541, −2.71754754913210593198617144376, −0.63538465993729824140218766919, 0.64312483239998709729013522268, 1.50625536995849469100973193574, 2.73712899957371409944983031997, 3.84810163845468062280256062921, 4.39126030807325844778158457796, 5.52863855598505950440559049984, 6.33456087453784565357568612116, 7.597505464941638234178837874523, 8.85385112632128639871979048981, 9.53231419331211355293542011971, 9.94888180380881496183814836974, 11.42290612363377359676931398127, 11.95065277356803595479166616876, 12.65066660487273645379342822305, 13.44064234570305783723536767212, 14.16538450582154722319995958915, 15.17569053512076617070696906605, 16.145414859437794641646036349500, 16.95162842895838746446016369058, 17.69578192678797472313729857855, 19.008157276339337195174812840405, 19.54044610489504442260637718520, 19.94270495886510988291485471141, 20.93474196573338269263278571771, 21.73830873832306333019221334430

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