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Properties

Label 1-177-177.11-r0-0-0
Degree $1$
Conductor $177$
Sign $0.718 - 0.695i$
Analytic cond. $0.821984$
Root an. cond. $0.821984$
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  + (0.907 − 0.419i)2-s + (0.647 − 0.762i)4-s + (0.856 + 0.515i)5-s + (0.0541 − 0.998i)7-s + (0.267 − 0.963i)8-s + (0.994 + 0.108i)10-s + (−0.161 + 0.986i)11-s + (−0.796 + 0.605i)13-s + (−0.370 − 0.928i)14-s + (−0.161 − 0.986i)16-s + (−0.0541 − 0.998i)17-s + (−0.725 + 0.687i)19-s + (0.947 − 0.319i)20-s + (0.267 + 0.963i)22-s + (0.976 − 0.214i)23-s + ⋯
L(s)  = 1  + (0.907 − 0.419i)2-s + (0.647 − 0.762i)4-s + (0.856 + 0.515i)5-s + (0.0541 − 0.998i)7-s + (0.267 − 0.963i)8-s + (0.994 + 0.108i)10-s + (−0.161 + 0.986i)11-s + (−0.796 + 0.605i)13-s + (−0.370 − 0.928i)14-s + (−0.161 − 0.986i)16-s + (−0.0541 − 0.998i)17-s + (−0.725 + 0.687i)19-s + (0.947 − 0.319i)20-s + (0.267 + 0.963i)22-s + (0.976 − 0.214i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.718 - 0.695i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.934941162 - 0.7830203586i\)
\(L(\frac12)\) \(\approx\) \(1.934941162 - 0.7830203586i\)
\(L(1)\) \(\approx\) \(1.762437694 - 0.4949045528i\)
\(L(1)\) \(\approx\) \(1.762437694 - 0.4949045528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.907 - 0.419i)T \)
5 \( 1 + (0.856 + 0.515i)T \)
7 \( 1 + (0.0541 - 0.998i)T \)
11 \( 1 + (-0.161 + 0.986i)T \)
13 \( 1 + (-0.796 + 0.605i)T \)
17 \( 1 + (-0.0541 - 0.998i)T \)
19 \( 1 + (-0.725 + 0.687i)T \)
23 \( 1 + (0.976 - 0.214i)T \)
29 \( 1 + (-0.907 - 0.419i)T \)
31 \( 1 + (0.725 + 0.687i)T \)
37 \( 1 + (-0.267 - 0.963i)T \)
41 \( 1 + (-0.976 - 0.214i)T \)
43 \( 1 + (0.161 + 0.986i)T \)
47 \( 1 + (-0.856 + 0.515i)T \)
53 \( 1 + (0.994 - 0.108i)T \)
61 \( 1 + (-0.907 + 0.419i)T \)
67 \( 1 + (-0.267 + 0.963i)T \)
71 \( 1 + (0.856 - 0.515i)T \)
73 \( 1 + (0.370 + 0.928i)T \)
79 \( 1 + (-0.947 + 0.319i)T \)
83 \( 1 + (-0.561 + 0.827i)T \)
89 \( 1 + (0.907 + 0.419i)T \)
97 \( 1 + (0.370 - 0.928i)T \)
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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

−27.55158165340914401164466681534, −26.1487455501548846689968667707, −25.388296028509000073764340332594, −24.46825923250056200835697176983, −24.01467390055595078851034236170, −22.52549703069148062158576705240, −21.64893142779167109390424139611, −21.27668165196851900161257170395, −20.01171748155917732352303847505, −18.74268551190002831880680546856, −17.33414162112258544477145845908, −16.801465709981760988281718773559, −15.430501543512763291536210152797, −14.80127895478588981096019464793, −13.44837029049423792891703497640, −12.87346518694931585383046375569, −11.814803986711137393393414251572, −10.55993882167621682879994883504, −8.99412507819692318960720191874, −8.160952533259531680201674028564, −6.538278223018669021860524868266, −5.620669265852727563996162546290, −4.87337941775146454883361460291, −3.1527504966005722179579461701, −2.04233379019527248648031688191, 1.66937121009434197159694576818, 2.77421742093859523030846326601, 4.241098771746111959440733284382, 5.207915717803979632725502216834, 6.67078510662514642683967891449, 7.28606858876855300330874530714, 9.54310887810339289523952161069, 10.24068600371212640108692464645, 11.22069147416130920740146489592, 12.487568874229861232465310268253, 13.44440306953872578985624988307, 14.31420779289553479555838438579, 14.99470008550635724641990900027, 16.492564914071246376152851425178, 17.461706240900446059441984919336, 18.69764488673199591070227507804, 19.74924981941905376727618620793, 20.79844634654844869555253882463, 21.36075779818583745505317697976, 22.762942295032385782504355030179, 22.96718551349777538374897141198, 24.34495618648493464908378554699, 25.18993723713275907529199761146, 26.2193421229464717505703151999, 27.301389131848110364480080186578

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