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Properties

Label 1-179-179.125-r0-0-0
Degree $1$
Conductor $179$
Sign $0.196 + 0.980i$
Analytic cond. $0.831272$
Root an. cond. $0.831272$
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  + (−0.458 + 0.888i)2-s + (0.362 + 0.932i)3-s + (−0.579 − 0.815i)4-s + (0.977 − 0.210i)5-s + (−0.994 − 0.105i)6-s + (−0.192 − 0.981i)7-s + (0.990 − 0.140i)8-s + (−0.737 + 0.675i)9-s + (−0.261 + 0.965i)10-s + (0.760 − 0.648i)11-s + (0.550 − 0.835i)12-s + (0.607 + 0.794i)13-s + (0.960 + 0.278i)14-s + (0.550 + 0.835i)15-s + (−0.329 + 0.944i)16-s + (0.844 + 0.535i)17-s + ⋯
L(s)  = 1  + (−0.458 + 0.888i)2-s + (0.362 + 0.932i)3-s + (−0.579 − 0.815i)4-s + (0.977 − 0.210i)5-s + (−0.994 − 0.105i)6-s + (−0.192 − 0.981i)7-s + (0.990 − 0.140i)8-s + (−0.737 + 0.675i)9-s + (−0.261 + 0.965i)10-s + (0.760 − 0.648i)11-s + (0.550 − 0.835i)12-s + (0.607 + 0.794i)13-s + (0.960 + 0.278i)14-s + (0.550 + 0.835i)15-s + (−0.329 + 0.944i)16-s + (0.844 + 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.196 + 0.980i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.196 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8932553437 + 0.7316539966i\)
\(L(\frac12)\) \(\approx\) \(0.8932553437 + 0.7316539966i\)
\(L(1)\) \(\approx\) \(0.9186589903 + 0.5329794957i\)
\(L(1)\) \(\approx\) \(0.9186589903 + 0.5329794957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.458 + 0.888i)T \)
3 \( 1 + (0.362 + 0.932i)T \)
5 \( 1 + (0.977 - 0.210i)T \)
7 \( 1 + (-0.192 - 0.981i)T \)
11 \( 1 + (0.760 - 0.648i)T \)
13 \( 1 + (0.607 + 0.794i)T \)
17 \( 1 + (0.844 + 0.535i)T \)
19 \( 1 + (-0.825 - 0.564i)T \)
23 \( 1 + (0.997 - 0.0705i)T \)
29 \( 1 + (-0.949 + 0.312i)T \)
31 \( 1 + (0.295 + 0.955i)T \)
37 \( 1 + (-0.949 - 0.312i)T \)
41 \( 1 + (-0.999 - 0.0352i)T \)
43 \( 1 + (0.911 + 0.411i)T \)
47 \( 1 + (-0.123 - 0.992i)T \)
53 \( 1 + (-0.896 + 0.442i)T \)
59 \( 1 + (0.662 + 0.749i)T \)
61 \( 1 + (-0.969 + 0.244i)T \)
67 \( 1 + (0.990 + 0.140i)T \)
71 \( 1 + (0.158 + 0.987i)T \)
73 \( 1 + (0.0176 - 0.999i)T \)
79 \( 1 + (-0.863 - 0.505i)T \)
83 \( 1 + (0.427 - 0.904i)T \)
89 \( 1 + (-0.458 - 0.888i)T \)
97 \( 1 + (0.489 - 0.871i)T \)
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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

−27.41891456634927379614452321431, −25.74551968993436834195643805493, −25.534549646838397294962509291275, −24.69744688676141435364737648601, −22.92879168389949875890158287173, −22.358246016299886866834280822954, −21.001839689044740431945787561937, −20.4884576023983751840461789748, −19.018492510672953712081338182738, −18.69086188520873301724575946522, −17.64758966953529100126663460640, −16.97093414577245650779365358623, −15.05433442889324616979023242802, −14.001269488118734353618308984912, −12.93260552756832876033819455505, −12.35591687362380896218801848919, −11.19603931391992520967943783522, −9.78448119879839431616124503732, −9.058354627875397873603492965691, −8.00028650568613877813101966158, −6.64063235715585939991879691623, −5.451189940280048494174742861, −3.3971731836778185737728419181, −2.35724355856289104907009541040, −1.39985511796797128430300921550, 1.44492201007543501665982057098, 3.59999927378035603607505782064, 4.75155106840732098432661728846, 5.9464983399336658805696225485, 6.96587857207549534208541884410, 8.59163788189299933817281262949, 9.158571089115143238475809010635, 10.23668864652267168447897425854, 10.96377523603920895920875098796, 13.2285115345677555165250869289, 14.04299647786264146006859130828, 14.67515097226641259522543416995, 16.046909686833499987479682733842, 16.907814283811777369361598752094, 17.201535858364722727749523820160, 18.85036456205216980621019522536, 19.70609006193400860840141269881, 20.91624104245763369789932308976, 21.74082632330878504863191534504, 22.85767501661414751023059905111, 23.862009449412836506035887298682, 24.993015484676394896384441443129, 25.85829068355012759912744869837, 26.35619755314328053792536796925, 27.36771461859364357564880587565

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