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Properties

Label 1-483-483.158-r0-0-0
Degree $1$
Conductor $483$
Sign $-0.187 + 0.982i$
Analytic cond. $2.24304$
Root an. cond. $2.24304$
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  + (−0.235 + 0.971i)2-s + (−0.888 − 0.458i)4-s + (0.928 + 0.371i)5-s + (0.654 − 0.755i)8-s + (−0.580 + 0.814i)10-s + (0.235 + 0.971i)11-s + (0.415 + 0.909i)13-s + (0.580 + 0.814i)16-s + (0.0475 − 0.998i)17-s + (−0.0475 − 0.998i)19-s + (−0.654 − 0.755i)20-s − 22-s + (0.723 + 0.690i)25-s + (−0.981 + 0.189i)26-s + (−0.841 − 0.540i)29-s + ⋯
L(s)  = 1  + (−0.235 + 0.971i)2-s + (−0.888 − 0.458i)4-s + (0.928 + 0.371i)5-s + (0.654 − 0.755i)8-s + (−0.580 + 0.814i)10-s + (0.235 + 0.971i)11-s + (0.415 + 0.909i)13-s + (0.580 + 0.814i)16-s + (0.0475 − 0.998i)17-s + (−0.0475 − 0.998i)19-s + (−0.654 − 0.755i)20-s − 22-s + (0.723 + 0.690i)25-s + (−0.981 + 0.189i)26-s + (−0.841 − 0.540i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8446312529 + 1.020811581i\)
\(L(\frac12)\) \(\approx\) \(0.8446312529 + 1.020811581i\)
\(L(1)\) \(\approx\) \(0.8995224214 + 0.5750821416i\)
\(L(1)\) \(\approx\) \(0.8995224214 + 0.5750821416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.235 + 0.971i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
11 \( 1 + (0.235 + 0.971i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.0475 - 0.998i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (0.786 + 0.618i)T \)
41 \( 1 + (0.142 + 0.989i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.995 + 0.0950i)T \)
59 \( 1 + (-0.580 + 0.814i)T \)
61 \( 1 + (0.327 + 0.945i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.888 - 0.458i)T \)
79 \( 1 + (0.995 + 0.0950i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−23.37927575799071460376375702614, −22.31931612283546088921156627416, −21.76377471768789933838502431022, −20.8554138098055218209569931914, −20.37831806045771761013947655889, −19.21120374662640980797547441432, −18.589364962429694725201143292394, −17.54575517816995097280529799735, −17.036487447133728890518622415392, −16.03284067779972132652467462665, −14.54558094461221488694954889472, −13.80196816028183714114499095957, −12.92873387724570512011251686330, −12.3451272915951084942578253300, −11.022111599492179335553395420063, −10.45459643539129825027295135081, −9.48958248774660904458422997170, −8.62628059475646484126625532874, −7.879444363406550079819110494818, −6.09273135650056837393287829421, −5.45853353822451414065675465593, −4.08416275713629708024945857505, −3.141433262927121720355554519817, −1.93711304689192184669394138843, −0.93677117747152821389281434588, 1.32269070959743984867250560814, 2.61670226393681092749589339935, 4.28057880447851402538841897202, 5.08688710684737649818173022392, 6.28389479963856766229735438757, 6.837928956994081122019965890622, 7.80337096851124353768993796839, 9.28488281479325673762029633841, 9.432065085342577873253382926611, 10.59761789645341535766771587949, 11.762382273081403607206070146283, 13.17450745696952515650706496240, 13.71617534909573784580970139938, 14.61227312650636732164556906206, 15.34666869520434975386183383981, 16.39178655142464110443298977865, 17.179154375633943286474506064144, 17.947759697421768307985141503352, 18.54030557606805574538980841116, 19.5513170716212698254527650113, 20.70762183118184199657437437619, 21.66368966649207892311983196031, 22.50922839372167555916959916513, 23.157678373135692294340917373005, 24.178202581377684577576776747816

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