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Properties

Label 1-179-179.57-r0-0-0
Degree $1$
Conductor $179$
Sign $-0.356 - 0.934i$
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  + (0.844 − 0.535i)2-s + (−0.261 + 0.965i)3-s + (0.427 − 0.904i)4-s + (−0.825 − 0.564i)5-s + (0.295 + 0.955i)6-s + (−0.688 − 0.725i)7-s + (−0.123 − 0.992i)8-s + (−0.863 − 0.505i)9-s + (−0.999 − 0.0352i)10-s + (−0.579 − 0.815i)11-s + (0.760 + 0.648i)12-s + (0.0176 − 0.999i)13-s + (−0.969 − 0.244i)14-s + (0.760 − 0.648i)15-s + (−0.635 − 0.772i)16-s + (0.880 + 0.474i)17-s + ⋯
L(s)  = 1  + (0.844 − 0.535i)2-s + (−0.261 + 0.965i)3-s + (0.427 − 0.904i)4-s + (−0.825 − 0.564i)5-s + (0.295 + 0.955i)6-s + (−0.688 − 0.725i)7-s + (−0.123 − 0.992i)8-s + (−0.863 − 0.505i)9-s + (−0.999 − 0.0352i)10-s + (−0.579 − 0.815i)11-s + (0.760 + 0.648i)12-s + (0.0176 − 0.999i)13-s + (−0.969 − 0.244i)14-s + (0.760 − 0.648i)15-s + (−0.635 − 0.772i)16-s + (0.880 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 179 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(179\)
Sign: $-0.356 - 0.934i$
Analytic conductor: \(0.831272\)
Root analytic conductor: \(0.831272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{179} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 179,\ (0:\ ),\ -0.356 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6301571598 - 0.9144647577i\)
\(L(\frac12)\) \(\approx\) \(0.6301571598 - 0.9144647577i\)
\(L(1)\) \(\approx\) \(1.021664299 - 0.4956146408i\)
\(L(1)\) \(\approx\) \(1.021664299 - 0.4956146408i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad179 \( 1 \)
good2 \( 1 + (0.844 - 0.535i)T \)
3 \( 1 + (-0.261 + 0.965i)T \)
5 \( 1 + (-0.825 - 0.564i)T \)
7 \( 1 + (-0.688 - 0.725i)T \)
11 \( 1 + (-0.579 - 0.815i)T \)
13 \( 1 + (0.0176 - 0.999i)T \)
17 \( 1 + (0.880 + 0.474i)T \)
19 \( 1 + (0.607 + 0.794i)T \)
23 \( 1 + (0.662 - 0.749i)T \)
29 \( 1 + (-0.783 + 0.621i)T \)
31 \( 1 + (-0.896 + 0.442i)T \)
37 \( 1 + (-0.783 - 0.621i)T \)
41 \( 1 + (0.911 + 0.411i)T \)
43 \( 1 + (0.362 - 0.932i)T \)
47 \( 1 + (0.0881 - 0.996i)T \)
53 \( 1 + (0.713 + 0.700i)T \)
59 \( 1 + (-0.737 - 0.675i)T \)
61 \( 1 + (-0.984 - 0.175i)T \)
67 \( 1 + (-0.123 + 0.992i)T \)
71 \( 1 + (-0.329 - 0.944i)T \)
73 \( 1 + (0.977 + 0.210i)T \)
79 \( 1 + (0.997 + 0.0705i)T \)
83 \( 1 + (0.550 - 0.835i)T \)
89 \( 1 + (0.844 + 0.535i)T \)
97 \( 1 + (0.990 - 0.140i)T \)
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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

−27.73158965941525116005213572602, −26.049025481263645766463926553785, −25.79071238286097608440622058537, −24.52331816599414827089614721502, −23.75955508686439599717915849311, −22.87648262929664414397699273192, −22.42479193302977345332947058411, −21.109487372397832111506454253678, −19.77594241993248812125883617677, −18.8537975463515193894473745846, −17.98758011912849032434138512561, −16.68533364260704800992305152790, −15.735394159541298669673686131668, −14.85384324862704145294622858110, −13.741833416365919632153138555017, −12.72392169795805902340325017740, −11.93845885326930777300877387268, −11.19982635891943490399299627801, −9.20304516154919106858231153926, −7.64771314699293426128422815529, −7.18176169492355086851824278458, −6.090403889321776024642508229948, −4.941990337329764676345027753585, −3.33332227917359695142826941515, −2.3226601132837582966805883557, 0.69347620064815732064262727668, 3.30243635026352891253549503587, 3.67308657553394785707839874862, 5.06333533320852034038827462038, 5.85217663446623126054670313020, 7.55787611525281197416756883449, 9.04987095224155199403050168102, 10.40347845813840466732045202947, 10.83154994116126575269811432798, 12.18324576342136206295348573133, 12.94303811598797241979198343383, 14.23596809518933378171623680512, 15.27716388849484737712531086977, 16.18365517545702950729733894925, 16.72880668685882648050959508742, 18.636700787569710450620941495313, 19.75921325677041271242469606938, 20.44865638998100588679731345542, 21.17779620887638941695735342934, 22.38827045747025921882063498907, 23.06961310498964503764150944278, 23.72167086545080178566989866917, 24.95003499169824556924941731365, 26.336050199435979778692601895485, 27.290414482294749848602326714220

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