Properties

Label 1-72-72.59-r0-0-0
Degree $1$
Conductor $72$
Sign $0.642 - 0.766i$
Analytic cond. $0.334366$
Root an. cond. $0.334366$
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)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 35-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s + 19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (0.5 + 0.866i)31-s − 35-s − 37-s + (0.5 + 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(72\)    =    \(2^{3} \cdot 3^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(0.334366\)
Root analytic conductor: \(0.334366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{72} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 72,\ (0:\ ),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8256783936 - 0.3850201581i\)
\(L(\frac12)\) \(\approx\) \(0.8256783936 - 0.3850201581i\)
\(L(1)\) \(\approx\) \(0.9657295487 - 0.2418462325i\)
\(L(1)\) \(\approx\) \(0.9657295487 - 0.2418462325i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 - T \)
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 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−31.47365284387212510527484027178, −30.76198118182756399963417571918, −29.86383860276153121854188414741, −28.29821303154952302982613649817, −27.5523666212794504881485136308, −26.36871616010120181541707939936, −25.26505887346781658518871709209, −24.20369078163529232283781407910, −22.74636448527918262607168535777, −22.18109369445376156327064564481, −20.683856857880770865253802663443, −19.55003745177711351360128864969, −18.295376777617149728141500040726, −17.579108694749484804602030326419, −15.58456159252951795433077305233, −15.118598841467356039227958215117, −13.706457356116864517578608271846, −12.083209241227550718772095990492, −11.22443392675669114780611968392, −9.78287036852542958536656612719, −8.27934023819982130331178300235, −7.04610367991045475940697044530, −5.56009399288233496114948471525, −3.84186622036355904314648850343, −2.24004668425538474525233672394, 1.280152197260028030800112306148, 3.74002024390602897682147946262, 4.860245271433104475227951032356, 6.65778239132726212866136137028, 8.13183863401421268741523653060, 9.14548552589160229265556507650, 10.910787912465851977167363821390, 11.85602292923144602115842235229, 13.35613524784961168721824939651, 14.30662366215205834290305926085, 16.0230132207712621025606396335, 16.69240992689120755000267012648, 18.02081219647978144864204048689, 19.503557143583352681058482142154, 20.34071796558502850726197051049, 21.41205749308946722549639921677, 22.83902782531965293166734826815, 24.10088757754853338482249620263, 24.495186011995033430334416316712, 26.32332905636736910326283304543, 27.08266425690759396433972013943, 28.23440745442363445517025022235, 29.24257735426601907009727017624, 30.52325296682130191117812107755, 31.41750544205144077923310704548

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