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Properties

Label 1-532-532.271-r0-0-0
Degree $1$
Conductor $532$
Sign $0.258 - 0.966i$
Analytic cond. $2.47059$
Root an. cond. $2.47059$
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  + (0.766 − 0.642i)3-s + (−0.766 + 0.642i)5-s + (0.173 − 0.984i)9-s − 11-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.766 + 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (−0.766 + 0.642i)5-s + (0.173 − 0.984i)9-s − 11-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (−0.173 − 0.984i)17-s + (0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + (0.766 + 0.642i)29-s + (−0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + (−0.5 − 0.866i)37-s + (0.5 − 0.866i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $0.258 - 0.966i$
Analytic conductor: \(2.47059\)
Root analytic conductor: \(2.47059\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{532} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 532,\ (0:\ ),\ 0.258 - 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.115863581 - 0.8566852363i\)
\(L(\frac12)\) \(\approx\) \(1.115863581 - 0.8566852363i\)
\(L(1)\) \(\approx\) \(1.117733221 - 0.3256481121i\)
\(L(1)\) \(\approx\) \(1.117733221 - 0.3256481121i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 - T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.766 + 0.642i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.173 - 0.984i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.766 + 0.642i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−23.60659899879096204552851870278, −22.96418934580864025542293897272, −21.66847607430966203266697910407, −20.99513936035064711923596519278, −20.45811070750197636775280797202, −19.40305586413261915298071704278, −18.99136154286280502916854599831, −17.69523431984182838459943986605, −16.57425493067790204804834099871, −15.82823402881528381956896547829, −15.374609974751297587141574867643, −14.34824743466229118832351607328, −13.278108091155044036811918358812, −12.723542870933245656161811348268, −11.35480775318593281424097167446, −10.66366658504441595205827844865, −9.60608680791297602867378499871, −8.509997908008664740217116359529, −8.23146639796295129938362123632, −7.05957723879162179294233690721, −5.560995717399586199702416634186, −4.60257959214471888248933186153, −3.79157038824654770968474076715, −2.829975992796347001768482728281, −1.41233338764767173534579984119, 0.729515659773404682606793901545, 2.37963380633375670217169195459, 3.10106384089495104953023112140, 4.073033829451347409466891158118, 5.48912236355318829041977459643, 6.778021117412832944292621824284, 7.403834027415974542028814424287, 8.26546184918019445299972224577, 9.05407783269338955967651120019, 10.37447686352469351856871748029, 11.17223939106169356588730590918, 12.18911900237067169215119816161, 13.09919157499745523954401236929, 13.83291827471431897278010314475, 14.789830717143410728113441006914, 15.52200232024830538373816097287, 16.23337590499356735291704752086, 17.78676370311332923406550294456, 18.44509769025222040906262096141, 18.92128301630056034569236164129, 19.9712154661225584217816140626, 20.570391192736394566841003840799, 21.45230311553051081315187705715, 22.81319215191263488156987149046, 23.25675418516403595868187866186

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