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Properties

Label 1-177-177.101-r0-0-0
Degree $1$
Conductor $177$
Sign $0.662 + 0.749i$
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  + (−0.370 − 0.928i)2-s + (−0.725 + 0.687i)4-s + (−0.647 − 0.762i)5-s + (−0.856 + 0.515i)7-s + (0.907 + 0.419i)8-s + (−0.468 + 0.883i)10-s + (0.0541 + 0.998i)11-s + (−0.976 − 0.214i)13-s + (0.796 + 0.605i)14-s + (0.0541 − 0.998i)16-s + (0.856 + 0.515i)17-s + (0.267 + 0.963i)19-s + (0.994 + 0.108i)20-s + (0.907 − 0.419i)22-s + (−0.561 + 0.827i)23-s + ⋯
L(s)  = 1  + (−0.370 − 0.928i)2-s + (−0.725 + 0.687i)4-s + (−0.647 − 0.762i)5-s + (−0.856 + 0.515i)7-s + (0.907 + 0.419i)8-s + (−0.468 + 0.883i)10-s + (0.0541 + 0.998i)11-s + (−0.976 − 0.214i)13-s + (0.796 + 0.605i)14-s + (0.0541 − 0.998i)16-s + (0.856 + 0.515i)17-s + (0.267 + 0.963i)19-s + (0.994 + 0.108i)20-s + (0.907 − 0.419i)22-s + (−0.561 + 0.827i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.662 + 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3861508902 + 0.1741081483i\)
\(L(\frac12)\) \(\approx\) \(0.3861508902 + 0.1741081483i\)
\(L(1)\) \(\approx\) \(0.5706092964 - 0.1160921514i\)
\(L(1)\) \(\approx\) \(0.5706092964 - 0.1160921514i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.370 - 0.928i)T \)
5 \( 1 + (-0.647 - 0.762i)T \)
7 \( 1 + (-0.856 + 0.515i)T \)
11 \( 1 + (0.0541 + 0.998i)T \)
13 \( 1 + (-0.976 - 0.214i)T \)
17 \( 1 + (0.856 + 0.515i)T \)
19 \( 1 + (0.267 + 0.963i)T \)
23 \( 1 + (-0.561 + 0.827i)T \)
29 \( 1 + (0.370 - 0.928i)T \)
31 \( 1 + (-0.267 + 0.963i)T \)
37 \( 1 + (-0.907 + 0.419i)T \)
41 \( 1 + (0.561 + 0.827i)T \)
43 \( 1 + (-0.0541 + 0.998i)T \)
47 \( 1 + (0.647 - 0.762i)T \)
53 \( 1 + (-0.468 - 0.883i)T \)
61 \( 1 + (0.370 + 0.928i)T \)
67 \( 1 + (-0.907 - 0.419i)T \)
71 \( 1 + (-0.647 + 0.762i)T \)
73 \( 1 + (-0.796 - 0.605i)T \)
79 \( 1 + (-0.994 - 0.108i)T \)
83 \( 1 + (-0.947 - 0.319i)T \)
89 \( 1 + (-0.370 + 0.928i)T \)
97 \( 1 + (-0.796 + 0.605i)T \)
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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

−26.914305450695328787848755764414, −26.403309511668170696698544004992, −25.56949367231774369902559105298, −24.27441021720311851078479984563, −23.638860266239069964711165812440, −22.52425034550744546532899077354, −22.029339099803706967855810342582, −20.0992188430013961733248341820, −19.1746757023590787262306298777, −18.65266259412843534430804724798, −17.323697105799669826557792367358, −16.3370584084497160357028739312, −15.69336023157836749741872382603, −14.433060298896427242005761265551, −13.81977822745318894583133920692, −12.36738553579876338647814623316, −10.91803882534873359967464101394, −9.99746771737473514119485189979, −8.86905214689726496222108128217, −7.51648493176658458859152571898, −6.924731020498757658080183382125, −5.74663661409556386326090500316, −4.24819699377371656855526981990, −2.975629541259456016785068179780, −0.3966383830828317670526767664, 1.589785146136245398613719005751, 3.09595135273686277556357077024, 4.224945519651233376211515748759, 5.45478730755421469418008938057, 7.38217012868244602509424515589, 8.32279072370887279149578797931, 9.576257485259325838491354012438, 10.12274498695456134888369045664, 11.930854877570758562087927973596, 12.24395603923896740799477149293, 13.14599821120227942025880880170, 14.68067733666093741122874493962, 15.9400550032522277975779505308, 16.88275354319727245928708368230, 17.87689381105787392974195864835, 19.18022360952312635262693975167, 19.64123668010278634233174059369, 20.60209806113955354246239358208, 21.587062631504620416820312605730, 22.67465424662985622878257637482, 23.35398887757918546113949618522, 24.86755382154709204661368867657, 25.689656990702039249713583145590, 26.85521967116756939693462599902, 27.76291377977594593120881752212

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