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Properties

Label 1-179-179.145-r0-0-0
Degree $1$
Conductor $179$
Sign $0.999 + 0.00738i$
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  + (−0.925 + 0.378i)2-s + (−0.458 + 0.888i)3-s + (0.713 − 0.700i)4-s + (−0.984 + 0.175i)5-s + (0.0881 − 0.996i)6-s + (0.911 − 0.411i)7-s + (−0.394 + 0.918i)8-s + (−0.579 − 0.815i)9-s + (0.844 − 0.535i)10-s + (−0.896 − 0.442i)11-s + (0.295 + 0.955i)12-s + (0.960 − 0.278i)13-s + (−0.688 + 0.725i)14-s + (0.295 − 0.955i)15-s + (0.0176 − 0.999i)16-s + (−0.0529 − 0.998i)17-s + ⋯
L(s)  = 1  + (−0.925 + 0.378i)2-s + (−0.458 + 0.888i)3-s + (0.713 − 0.700i)4-s + (−0.984 + 0.175i)5-s + (0.0881 − 0.996i)6-s + (0.911 − 0.411i)7-s + (−0.394 + 0.918i)8-s + (−0.579 − 0.815i)9-s + (0.844 − 0.535i)10-s + (−0.896 − 0.442i)11-s + (0.295 + 0.955i)12-s + (0.960 − 0.278i)13-s + (−0.688 + 0.725i)14-s + (0.295 − 0.955i)15-s + (0.0176 − 0.999i)16-s + (−0.0529 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.999 + 0.00738i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.999 + 0.00738i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5159653256 + 0.001904499784i\)
\(L(\frac12)\) \(\approx\) \(0.5159653256 + 0.001904499784i\)
\(L(1)\) \(\approx\) \(0.5493696961 + 0.1108842537i\)
\(L(1)\) \(\approx\) \(0.5493696961 + 0.1108842537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.925 + 0.378i)T \)
3 \( 1 + (-0.458 + 0.888i)T \)
5 \( 1 + (-0.984 + 0.175i)T \)
7 \( 1 + (0.911 - 0.411i)T \)
11 \( 1 + (-0.896 - 0.442i)T \)
13 \( 1 + (0.960 - 0.278i)T \)
17 \( 1 + (-0.0529 - 0.998i)T \)
19 \( 1 + (-0.520 - 0.854i)T \)
23 \( 1 + (0.550 + 0.835i)T \)
29 \( 1 + (-0.261 - 0.965i)T \)
31 \( 1 + (0.489 + 0.871i)T \)
37 \( 1 + (-0.261 + 0.965i)T \)
41 \( 1 + (0.880 - 0.474i)T \)
43 \( 1 + (0.938 + 0.345i)T \)
47 \( 1 + (0.158 - 0.987i)T \)
53 \( 1 + (0.990 + 0.140i)T \)
59 \( 1 + (0.760 + 0.648i)T \)
61 \( 1 + (-0.949 - 0.312i)T \)
67 \( 1 + (-0.394 - 0.918i)T \)
71 \( 1 + (0.607 - 0.794i)T \)
73 \( 1 + (-0.969 + 0.244i)T \)
79 \( 1 + (0.427 - 0.904i)T \)
83 \( 1 + (-0.994 - 0.105i)T \)
89 \( 1 + (-0.925 - 0.378i)T \)
97 \( 1 + (-0.635 + 0.772i)T \)
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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

−27.78140575219285552025887695516, −26.47876286769548475591692671366, −25.511453280922329623424393717158, −24.41900526413179273806004655968, −23.74112307913216329485198827825, −22.73577232024212208908842633432, −21.20381079651936397983167067235, −20.48237820148313767707472755337, −19.252004637031596228464438057138, −18.626264289171699896623662187473, −17.87236315885655850449862477541, −16.822214602565501383072967767674, −15.859857135125088310050674097791, −14.70232349563928915716906699572, −12.866027939646696183894769788961, −12.31998918053953321032091145181, −11.16851338346002693544320114736, −10.69497604492268853367491306413, −8.66317621205008915255490101699, −8.10785492595600794017185167562, −7.21663451000851230489798105324, −5.84841563932120139952189995295, −4.18869196683597105559963180625, −2.43199512973177429849234861518, −1.23269269773612966044907017397, 0.697804521056833853304617327695, 2.9891338136409374542896439204, 4.52440750810009634608686031297, 5.58083384277261078078665617609, 7.046667694767019189892794062015, 8.10944910346923402250982821908, 8.974405524838195716735288641710, 10.45306917748690389027231734209, 11.068498455673840238527398900119, 11.72373936674142779019136657262, 13.78314785719559839843788965498, 15.17834203031148719865340249992, 15.609573825256779838951723633817, 16.5071070965092865556593465536, 17.59122496613974534753250456807, 18.369880359747978321798249652078, 19.565772494959889626769035614949, 20.63478676804679343837279042952, 21.20362660834349800712262733978, 22.952922708076969639288721550368, 23.49123066835228476002322752854, 24.39223767137045348773757450540, 25.87993554284737595099971028440, 26.603284419126975178784092083256, 27.38473211848147203684315057773

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