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Properties

Label 1-483-483.137-r0-0-0
Degree $1$
Conductor $483$
Sign $-0.895 - 0.444i$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s − 22-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s − 29-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s − 22-s + (−0.5 − 0.866i)25-s + (0.5 − 0.866i)26-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2378563020 - 1.015346201i\)
\(L(\frac12)\) \(\approx\) \(0.2378563020 - 1.015346201i\)
\(L(1)\) \(\approx\) \(0.8417075447 - 0.5918830088i\)
\(L(1)\) \(\approx\) \(0.8417075447 - 0.5918830088i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 - T \)
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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

−23.87053198625308777420389711691, −23.55579801850331624632950613772, −22.68273414564915325479641649698, −21.72213853120684204781066509362, −20.66922696667498813250163691800, −20.270174467089902619142581760305, −18.83178037921097621789562565175, −17.98300416512620991021665741464, −17.047976778485056035060454864336, −16.29850462530126333451625118113, −15.53296030589053522982682800925, −14.88472791390639459708148899166, −13.67398751257198849330141049250, −12.9043557054305835101613585261, −12.29201141230708962979359437209, −11.22455999143780600745969382687, −9.83435958378938570458578941043, −8.72721758189174948473670631715, −8.095064871713301511147780962632, −7.18793975297992429982628129554, −6.03439936325565563479851509498, −5.13161624646600089124902142921, −4.22089071338530559257992513705, −3.38084487509685521685069178020, −1.60248067500269205504666717004, 0.48679260992720295638304431382, 2.17014165745857022018904864753, 3.16786492845561141358996646716, 3.86447141735159348808859129667, 5.14532854316067117017893444293, 6.11759483150464553220200864460, 7.17928929107842462836388065165, 8.44450059095020487463761651142, 9.42884618359316600782583426112, 10.56544117523398317869050647232, 11.26706094478374630346400650632, 11.700093131506333368539528063232, 13.22294001573360701952527531608, 13.584491963097275356254133610220, 14.66775225188578775729576205650, 15.48295205332703937236438181775, 16.28324917105851673281591240938, 17.93895054468268724280310936788, 18.4657795769870503808827138725, 19.14533401183502697113574171196, 20.13418358752685070969768578160, 20.80400664381882297254544147205, 21.938125208587024834975183656181, 22.310900397615858199244804066054, 23.38684281854981655985185764110

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