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Properties

Label 1-177-177.62-r1-0-0
Degree $1$
Conductor $177$
Sign $-0.979 - 0.199i$
Analytic cond. $19.0212$
Root an. cond. $19.0212$
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  + (−0.647 + 0.762i)2-s + (−0.161 − 0.986i)4-s + (−0.468 − 0.883i)5-s + (−0.994 − 0.108i)7-s + (0.856 + 0.515i)8-s + (0.976 + 0.214i)10-s + (0.947 + 0.319i)11-s + (0.267 − 0.963i)13-s + (0.725 − 0.687i)14-s + (−0.947 + 0.319i)16-s + (0.994 − 0.108i)17-s + (0.0541 − 0.998i)19-s + (−0.796 + 0.605i)20-s + (−0.856 + 0.515i)22-s + (−0.907 + 0.419i)23-s + ⋯
L(s)  = 1  + (−0.647 + 0.762i)2-s + (−0.161 − 0.986i)4-s + (−0.468 − 0.883i)5-s + (−0.994 − 0.108i)7-s + (0.856 + 0.515i)8-s + (0.976 + 0.214i)10-s + (0.947 + 0.319i)11-s + (0.267 − 0.963i)13-s + (0.725 − 0.687i)14-s + (−0.947 + 0.319i)16-s + (0.994 − 0.108i)17-s + (0.0541 − 0.998i)19-s + (−0.796 + 0.605i)20-s + (−0.856 + 0.515i)22-s + (−0.907 + 0.419i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.979 - 0.199i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ -0.979 - 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01137376235 - 0.1128989807i\)
\(L(\frac12)\) \(\approx\) \(0.01137376235 - 0.1128989807i\)
\(L(1)\) \(\approx\) \(0.5581301345 + 0.02638318630i\)
\(L(1)\) \(\approx\) \(0.5581301345 + 0.02638318630i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.647 + 0.762i)T \)
5 \( 1 + (-0.468 - 0.883i)T \)
7 \( 1 + (-0.994 - 0.108i)T \)
11 \( 1 + (0.947 + 0.319i)T \)
13 \( 1 + (0.267 - 0.963i)T \)
17 \( 1 + (0.994 - 0.108i)T \)
19 \( 1 + (0.0541 - 0.998i)T \)
23 \( 1 + (-0.907 + 0.419i)T \)
29 \( 1 + (-0.647 - 0.762i)T \)
31 \( 1 + (0.0541 + 0.998i)T \)
37 \( 1 + (-0.856 + 0.515i)T \)
41 \( 1 + (-0.907 - 0.419i)T \)
43 \( 1 + (-0.947 + 0.319i)T \)
47 \( 1 + (-0.468 + 0.883i)T \)
53 \( 1 + (-0.976 + 0.214i)T \)
61 \( 1 + (0.647 - 0.762i)T \)
67 \( 1 + (-0.856 - 0.515i)T \)
71 \( 1 + (-0.468 + 0.883i)T \)
73 \( 1 + (-0.725 + 0.687i)T \)
79 \( 1 + (0.796 - 0.605i)T \)
83 \( 1 + (0.370 + 0.928i)T \)
89 \( 1 + (-0.647 - 0.762i)T \)
97 \( 1 + (-0.725 - 0.687i)T \)
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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

−27.642040891274637311778339579141, −26.6648187909029968338672023059, −25.99035631888775825225312659518, −25.134227801493914064659315784051, −23.533481045150523362606042252557, −22.44113879185206495794784168869, −21.961249501142658998353009039322, −20.70169550148765609087780685341, −19.55534737278808954071594887831, −18.946562236268503806151872382276, −18.30301721833526161779264014029, −16.7337190745727884608552379059, −16.248762828184511724075641199063, −14.684266202218726580428012068335, −13.64434119895828021279094243052, −12.22217393844077905409635077211, −11.64372860252626032546743885434, −10.39916809547561752474393167018, −9.600439299622996266541299765879, −8.43012770576434741795988203320, −7.18214502820028404404433659739, −6.2121204536264285356504831209, −3.89085270781697205185705716322, −3.3340460476392254282940003359, −1.775931005353593691831981212702, 0.05501017614173117919929667776, 1.300043994661242414373750345768, 3.53426158893934816715879356294, 4.96412893043984369435540015169, 6.1059772274621477063554303894, 7.25643557655887516137317885712, 8.31871467179930407992143669109, 9.33864065014204393182931044342, 10.13237698587079757078318058075, 11.66628597562896643673309406128, 12.82110606259469911992125110365, 13.89955468415517334982484658727, 15.27370170868221030147576246947, 15.94969659042716391689925669144, 16.86072517749929801050785897561, 17.64519141893825870678430035768, 19.020693087841170263800703087304, 19.7540922609582151623287952005, 20.46894687535546667293711309185, 22.22902903654284183395396277465, 23.10142415988418538589496668690, 23.94463587517835250866996681466, 25.03417062200294695439315610149, 25.56655279613902350524720054543, 26.692763570402144059126385573465

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