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Properties

Label 1-179-179.101-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.228 - 0.973i$
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  + (0.0176 − 0.999i)2-s + (−0.329 + 0.944i)3-s + (−0.999 − 0.0352i)4-s + (0.760 − 0.648i)5-s + (0.938 + 0.345i)6-s + (−0.123 − 0.992i)7-s + (−0.0529 + 0.998i)8-s + (−0.783 − 0.621i)9-s + (−0.635 − 0.772i)10-s + (−0.261 + 0.965i)11-s + (0.362 − 0.932i)12-s + (0.427 − 0.904i)13-s + (−0.994 + 0.105i)14-s + (0.362 + 0.932i)15-s + (0.997 + 0.0705i)16-s + (0.977 − 0.210i)17-s + ⋯
L(s)  = 1  + (0.0176 − 0.999i)2-s + (−0.329 + 0.944i)3-s + (−0.999 − 0.0352i)4-s + (0.760 − 0.648i)5-s + (0.938 + 0.345i)6-s + (−0.123 − 0.992i)7-s + (−0.0529 + 0.998i)8-s + (−0.783 − 0.621i)9-s + (−0.635 − 0.772i)10-s + (−0.261 + 0.965i)11-s + (0.362 − 0.932i)12-s + (0.427 − 0.904i)13-s + (−0.994 + 0.105i)14-s + (0.362 + 0.932i)15-s + (0.997 + 0.0705i)16-s + (0.977 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.228 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.228 - 0.973i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.228 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5828412279 - 0.7354320192i\)
\(L(\frac12)\) \(\approx\) \(0.5828412279 - 0.7354320192i\)
\(L(1)\) \(\approx\) \(0.8072452136 - 0.4549048881i\)
\(L(1)\) \(\approx\) \(0.8072452136 - 0.4549048881i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.0176 - 0.999i)T \)
3 \( 1 + (-0.329 + 0.944i)T \)
5 \( 1 + (0.760 - 0.648i)T \)
7 \( 1 + (-0.123 - 0.992i)T \)
11 \( 1 + (-0.261 + 0.965i)T \)
13 \( 1 + (0.427 - 0.904i)T \)
17 \( 1 + (0.977 - 0.210i)T \)
19 \( 1 + (-0.579 - 0.815i)T \)
23 \( 1 + (-0.688 - 0.725i)T \)
29 \( 1 + (0.489 - 0.871i)T \)
31 \( 1 + (-0.458 - 0.888i)T \)
37 \( 1 + (0.489 + 0.871i)T \)
41 \( 1 + (-0.394 - 0.918i)T \)
43 \( 1 + (0.158 + 0.987i)T \)
47 \( 1 + (0.804 + 0.593i)T \)
53 \( 1 + (0.844 + 0.535i)T \)
59 \( 1 + (-0.949 + 0.312i)T \)
61 \( 1 + (0.295 + 0.955i)T \)
67 \( 1 + (-0.0529 - 0.998i)T \)
71 \( 1 + (-0.863 + 0.505i)T \)
73 \( 1 + (0.550 - 0.835i)T \)
79 \( 1 + (-0.192 + 0.981i)T \)
83 \( 1 + (0.911 + 0.411i)T \)
89 \( 1 + (0.0176 + 0.999i)T \)
97 \( 1 + (-0.925 + 0.378i)T \)
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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

−27.56998848121041171742748112042, −26.28207448326906682034590621946, −25.41320756976656360342787509896, −24.9863644442212011395309318397, −23.77763460444103902309678881540, −23.19865439583489845062672796849, −21.863993903638663975101451547082, −21.50864752810328593926226778084, −19.20664519933880438574024282556, −18.5977421983608576410684007226, −18.04021558047519273205848581489, −16.86490988495541446786827759142, −16.05694630398337919328707461602, −14.5549417657723650709349205005, −13.98567163380901819060568114425, −12.96336577554338833295804336196, −11.896803515730624838779497100117, −10.47936523128614067846730415853, −9.06763695355581439096964750993, −8.1504880795611822109129285246, −6.89260273661397744641066946581, −5.96076565100022547368049736358, −5.49861840615204067585929481781, −3.321262324001824954615129035292, −1.71824585367994124387729962400, 0.838910378652298462055807280170, 2.608407198210166900938676307319, 4.06694576173869683776307283571, 4.83647803928109946341880859694, 5.981383285056572416639718826422, 8.02907102831600520711727088655, 9.33266992871091977890768599815, 10.12016148007209657165921184537, 10.67243069808359483720727981719, 12.075063367218681409925790642258, 13.04699863620531015462919948952, 13.986402082918535436757910580866, 15.18769787047193830350799334998, 16.62897911864772108995789433313, 17.340033072754107496580723628627, 18.13750298898179047298532568372, 19.82822382514346880193393449464, 20.57487549632455724395743575348, 20.97931953214707335183155717365, 22.1688323969248378705388390857, 22.945021654446802879055389985225, 23.78764791517877163946231066781, 25.58835277693635352938731620666, 26.22416189008521154726333760738, 27.48678674506550069924573610495

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