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Properties

Label 1-179-179.158-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.820 + 0.571i$
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  + (0.911 + 0.411i)2-s + (−0.192 + 0.981i)3-s + (0.662 + 0.749i)4-s + (−0.329 + 0.944i)5-s + (−0.579 + 0.815i)6-s + (−0.984 − 0.175i)7-s + (0.295 + 0.955i)8-s + (−0.925 − 0.378i)9-s + (−0.688 + 0.725i)10-s + (0.997 + 0.0705i)11-s + (−0.863 + 0.505i)12-s + (−0.394 − 0.918i)13-s + (−0.825 − 0.564i)14-s + (−0.863 − 0.505i)15-s + (−0.123 + 0.992i)16-s + (0.362 + 0.932i)17-s + ⋯
L(s)  = 1  + (0.911 + 0.411i)2-s + (−0.192 + 0.981i)3-s + (0.662 + 0.749i)4-s + (−0.329 + 0.944i)5-s + (−0.579 + 0.815i)6-s + (−0.984 − 0.175i)7-s + (0.295 + 0.955i)8-s + (−0.925 − 0.378i)9-s + (−0.688 + 0.725i)10-s + (0.997 + 0.0705i)11-s + (−0.863 + 0.505i)12-s + (−0.394 − 0.918i)13-s + (−0.825 − 0.564i)14-s + (−0.863 − 0.505i)15-s + (−0.123 + 0.992i)16-s + (0.362 + 0.932i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4512820062 + 1.437954521i\)
\(L(\frac12)\) \(\approx\) \(0.4512820062 + 1.437954521i\)
\(L(1)\) \(\approx\) \(0.9981457614 + 0.9935791858i\)
\(L(1)\) \(\approx\) \(0.9981457614 + 0.9935791858i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.911 + 0.411i)T \)
3 \( 1 + (-0.192 + 0.981i)T \)
5 \( 1 + (-0.329 + 0.944i)T \)
7 \( 1 + (-0.984 - 0.175i)T \)
11 \( 1 + (0.997 + 0.0705i)T \)
13 \( 1 + (-0.394 - 0.918i)T \)
17 \( 1 + (0.362 + 0.932i)T \)
19 \( 1 + (-0.635 - 0.772i)T \)
23 \( 1 + (0.804 + 0.593i)T \)
29 \( 1 + (0.960 - 0.278i)T \)
31 \( 1 + (0.427 + 0.904i)T \)
37 \( 1 + (0.960 + 0.278i)T \)
41 \( 1 + (-0.949 + 0.312i)T \)
43 \( 1 + (-0.783 + 0.621i)T \)
47 \( 1 + (-0.896 + 0.442i)T \)
53 \( 1 + (0.550 + 0.835i)T \)
59 \( 1 + (0.227 - 0.973i)T \)
61 \( 1 + (0.607 - 0.794i)T \)
67 \( 1 + (0.295 - 0.955i)T \)
71 \( 1 + (0.990 - 0.140i)T \)
73 \( 1 + (0.158 + 0.987i)T \)
79 \( 1 + (-0.0529 - 0.998i)T \)
83 \( 1 + (-0.737 - 0.675i)T \)
89 \( 1 + (0.911 - 0.411i)T \)
97 \( 1 + (-0.994 - 0.105i)T \)
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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

−27.24611667780315265343648342690, −25.35710056127202275155767603240, −24.91375402394614691670023613757, −23.980372097849878888683259179843, −23.11497749972617888747227610721, −22.43044297230729574349646776205, −21.20127587743345568105225856966, −20.05542387438952600486558406352, −19.375317883039863319287905516698, −18.67964740274450108631145296557, −16.79945520496964291611127490143, −16.396026496087191245724252997634, −14.85380591608991018873491713009, −13.78279240301449107904542043241, −12.921790163735681162416452216408, −12.0767962111154301659717276765, −11.59745949175464351375647743607, −9.84411183137594110053555923334, −8.69044279149536005080273919404, −7.05370770693503274673913522362, −6.30067198773021443689979179451, −5.07095988340524771233857528909, −3.801201648739725047782438670122, −2.3574710257804776176213254277, −0.96611517036954088524444927977, 2.93918377762172709061560537569, 3.54957728181078285705543685499, 4.71224579614543647904121549837, 6.14475875345602375032763764256, 6.79711648203881266132730026473, 8.29163384119283358396855972177, 9.82373482537431936132933381848, 10.78720312413514818378974316899, 11.77605927634750449167545774579, 12.9097435452550965250139818322, 14.21792956248391334086865883981, 15.08845118105734725571618822580, 15.607113489961517656075536473768, 16.80996159074493695092227176335, 17.51804644619299902955948948767, 19.46627092895373696559975134244, 19.99216925934325988526124550510, 21.621268411127929839677940702902, 21.93108725244858336926021300499, 23.019372070615352023586657751366, 23.30512853216417510984204229240, 25.089795489406876147945803012361, 25.76135914056427543695739193721, 26.639668240159389735706767756899, 27.47875316604343769568755902882

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