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Properties

Label 1-179-179.147-r0-0-0
Degree $1$
Conductor $179$
Sign $0.997 + 0.0723i$
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  + (−0.984 − 0.175i)2-s + (0.977 + 0.210i)3-s + (0.938 + 0.345i)4-s + (0.713 − 0.700i)5-s + (−0.925 − 0.378i)6-s + (−0.329 + 0.944i)7-s + (−0.863 − 0.505i)8-s + (0.911 + 0.411i)9-s + (−0.825 + 0.564i)10-s + (0.880 − 0.474i)11-s + (0.844 + 0.535i)12-s + (0.295 + 0.955i)13-s + (0.489 − 0.871i)14-s + (0.844 − 0.535i)15-s + (0.760 + 0.648i)16-s + (−0.520 − 0.854i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.175i)2-s + (0.977 + 0.210i)3-s + (0.938 + 0.345i)4-s + (0.713 − 0.700i)5-s + (−0.925 − 0.378i)6-s + (−0.329 + 0.944i)7-s + (−0.863 − 0.505i)8-s + (0.911 + 0.411i)9-s + (−0.825 + 0.564i)10-s + (0.880 − 0.474i)11-s + (0.844 + 0.535i)12-s + (0.295 + 0.955i)13-s + (0.489 − 0.871i)14-s + (0.844 − 0.535i)15-s + (0.760 + 0.648i)16-s + (−0.520 − 0.854i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $0.997 + 0.0723i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ 0.997 + 0.0723i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.163040452 + 0.04215047094i\)
\(L(\frac12)\) \(\approx\) \(1.163040452 + 0.04215047094i\)
\(L(1)\) \(\approx\) \(1.059224902 + 0.003577138986i\)
\(L(1)\) \(\approx\) \(1.059224902 + 0.003577138986i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (-0.984 - 0.175i)T \)
3 \( 1 + (0.977 + 0.210i)T \)
5 \( 1 + (0.713 - 0.700i)T \)
7 \( 1 + (-0.329 + 0.944i)T \)
11 \( 1 + (0.880 - 0.474i)T \)
13 \( 1 + (0.295 + 0.955i)T \)
17 \( 1 + (-0.520 - 0.854i)T \)
19 \( 1 + (-0.994 - 0.105i)T \)
23 \( 1 + (-0.261 + 0.965i)T \)
29 \( 1 + (-0.394 + 0.918i)T \)
31 \( 1 + (-0.0529 - 0.998i)T \)
37 \( 1 + (-0.394 - 0.918i)T \)
41 \( 1 + (0.607 - 0.794i)T \)
43 \( 1 + (0.0176 + 0.999i)T \)
47 \( 1 + (0.997 + 0.0705i)T \)
53 \( 1 + (0.804 - 0.593i)T \)
59 \( 1 + (-0.999 + 0.0352i)T \)
61 \( 1 + (0.990 + 0.140i)T \)
67 \( 1 + (-0.863 + 0.505i)T \)
71 \( 1 + (0.550 + 0.835i)T \)
73 \( 1 + (-0.896 + 0.442i)T \)
79 \( 1 + (0.362 - 0.932i)T \)
83 \( 1 + (-0.458 - 0.888i)T \)
89 \( 1 + (-0.984 + 0.175i)T \)
97 \( 1 + (-0.737 + 0.675i)T \)
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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

−26.96789083793749145898720658130, −26.35340362795762937709173078692, −25.51187578891896019643018926441, −25.00430928559535460846853864805, −23.81273355411655920048615518374, −22.55218824783341275961199878291, −21.20319261927863916323256675804, −20.24327137851263425179437926060, −19.57877549959967759655264523717, −18.64437480897214220943759817991, −17.61835171661335823315365392112, −16.911038421056962221300488008630, −15.37721535277635081747669670939, −14.69323888679664778006966946157, −13.667295996684402294700428275966, −12.51350279411241311228612595900, −10.67605959818715055057754355705, −10.1801840170762234275528891822, −9.09165440776795434891849428265, −8.04165288480487752392107076721, −6.90294848330370882612179865587, −6.27399197113210696515844259584, −3.91415144365293696923305628572, −2.60024295741078685074973130159, −1.44340546268911438865473125837, 1.62746993044423672200107940376, 2.527863111637639186189644050540, 4.017783241938502194584907703438, 5.85165799263996873610276764207, 7.045068916324154734220081230769, 8.6161892512426861343766322174, 9.0638621719573292636465503369, 9.67756421692193571169670962447, 11.19938680357998368582046515559, 12.37317121557064239605588255436, 13.46624795795263484833109899076, 14.64727325293691556807975353132, 15.86771545191459459801042201289, 16.51717139366371021796325695428, 17.73449381054830326712728522605, 18.831066194842995921620160844195, 19.49885753433995785736268582059, 20.49644108158838789521855476738, 21.40777896617328482083201178838, 21.93238355488163057290662645239, 24.19066230732562864103908467860, 24.80076810687072655185214967078, 25.59452301128685281888127349147, 26.20693120987981305779875936097, 27.52161866549460423387104766020

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