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Properties

Label 1-177-177.38-r0-0-0
Degree $1$
Conductor $177$
Sign $-0.937 + 0.348i$
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  + (0.468 + 0.883i)2-s + (−0.561 + 0.827i)4-s + (−0.976 − 0.214i)5-s + (0.796 + 0.605i)7-s + (−0.994 − 0.108i)8-s + (−0.267 − 0.963i)10-s + (−0.370 + 0.928i)11-s + (−0.0541 + 0.998i)13-s + (−0.161 + 0.986i)14-s + (−0.370 − 0.928i)16-s + (−0.796 + 0.605i)17-s + (−0.947 − 0.319i)19-s + (0.725 − 0.687i)20-s + (−0.994 + 0.108i)22-s + (−0.856 − 0.515i)23-s + ⋯
L(s)  = 1  + (0.468 + 0.883i)2-s + (−0.561 + 0.827i)4-s + (−0.976 − 0.214i)5-s + (0.796 + 0.605i)7-s + (−0.994 − 0.108i)8-s + (−0.267 − 0.963i)10-s + (−0.370 + 0.928i)11-s + (−0.0541 + 0.998i)13-s + (−0.161 + 0.986i)14-s + (−0.370 − 0.928i)16-s + (−0.796 + 0.605i)17-s + (−0.947 − 0.319i)19-s + (0.725 − 0.687i)20-s + (−0.994 + 0.108i)22-s + (−0.856 − 0.515i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.937 + 0.348i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ -0.937 + 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1672984212 + 0.9300405565i\)
\(L(\frac12)\) \(\approx\) \(0.1672984212 + 0.9300405565i\)
\(L(1)\) \(\approx\) \(0.7222549466 + 0.6795618993i\)
\(L(1)\) \(\approx\) \(0.7222549466 + 0.6795618993i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.468 + 0.883i)T \)
5 \( 1 + (-0.976 - 0.214i)T \)
7 \( 1 + (0.796 + 0.605i)T \)
11 \( 1 + (-0.370 + 0.928i)T \)
13 \( 1 + (-0.0541 + 0.998i)T \)
17 \( 1 + (-0.796 + 0.605i)T \)
19 \( 1 + (-0.947 - 0.319i)T \)
23 \( 1 + (-0.856 - 0.515i)T \)
29 \( 1 + (-0.468 + 0.883i)T \)
31 \( 1 + (0.947 - 0.319i)T \)
37 \( 1 + (0.994 - 0.108i)T \)
41 \( 1 + (0.856 - 0.515i)T \)
43 \( 1 + (0.370 + 0.928i)T \)
47 \( 1 + (0.976 - 0.214i)T \)
53 \( 1 + (-0.267 + 0.963i)T \)
61 \( 1 + (-0.468 - 0.883i)T \)
67 \( 1 + (0.994 + 0.108i)T \)
71 \( 1 + (-0.976 + 0.214i)T \)
73 \( 1 + (0.161 - 0.986i)T \)
79 \( 1 + (-0.725 + 0.687i)T \)
83 \( 1 + (0.647 + 0.762i)T \)
89 \( 1 + (0.468 - 0.883i)T \)
97 \( 1 + (0.161 + 0.986i)T \)
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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

−27.20784300779009823277571447600, −26.47788004633462599571878031408, −24.6542917954533856473248718028, −23.823756581943287190013560310505, −23.109472648083266448480845605931, −22.1718654958896750108868874061, −21.05584426309845797384056707185, −20.23099349565490357535790907006, −19.42384829575957624699281773644, −18.45097942135708636827086016408, −17.44800459836977108706554060729, −15.87778974462091108004138817899, −14.97585348524622134406825610857, −13.94164426090613979494590458037, −13.00430101533392344414005004900, −11.74122233043769036320506121932, −11.0510348880606042315631918690, −10.239810266274586716542234949977, −8.58167586671849238673698392393, −7.684881109059492218025215553761, −6.00177460650511119209722572030, −4.65485580800450352167715147650, −3.76281265260749396612848335545, −2.50365162845834394392936490452, −0.65796839953997695868200103743, 2.31650495215218853437684197144, 4.2259025285902236334210971993, 4.656659836387811700503236152696, 6.190343672563472587838184896133, 7.36838031397027904968172607820, 8.28574022625298786399823204920, 9.14689325415618225526933975055, 11.03884935535686085008356885441, 12.11978369007146522378566560863, 12.83839698460654774850241718146, 14.3119980149386991075602849440, 15.10582416305791382925017609129, 15.79281032338766687938288985498, 16.92338421031197533852071692758, 17.90862861494462821845619246864, 18.89529457447092696507604380309, 20.22489303079183194245461957696, 21.29064835325020213443018183706, 22.17629342287381051971919272491, 23.34328136415150993782942811263, 23.95788579350233561875685110727, 24.68688082764730388242466741071, 25.93151168735938802208465137168, 26.63263785113586415804499218146, 27.80377962482967719104203566

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