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Properties

Label 1-179-179.17-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: $\chi_{179} (17, \cdot )$
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

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

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