Properties

Label 1-177-177.140-r1-0-0
Degree $1$
Conductor $177$
Sign $0.999 + 0.0392i$
Analytic cond. $19.0212$
Root an. cond. $19.0212$
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  + (0.947 − 0.319i)2-s + (0.796 − 0.605i)4-s + (0.370 + 0.928i)5-s + (0.907 + 0.419i)7-s + (0.561 − 0.827i)8-s + (0.647 + 0.762i)10-s + (−0.267 − 0.963i)11-s + (0.468 + 0.883i)13-s + (0.994 + 0.108i)14-s + (0.267 − 0.963i)16-s + (−0.907 + 0.419i)17-s + (0.976 + 0.214i)19-s + (0.856 + 0.515i)20-s + (−0.561 − 0.827i)22-s + (0.161 + 0.986i)23-s + ⋯
L(s)  = 1  + (0.947 − 0.319i)2-s + (0.796 − 0.605i)4-s + (0.370 + 0.928i)5-s + (0.907 + 0.419i)7-s + (0.561 − 0.827i)8-s + (0.647 + 0.762i)10-s + (−0.267 − 0.963i)11-s + (0.468 + 0.883i)13-s + (0.994 + 0.108i)14-s + (0.267 − 0.963i)16-s + (−0.907 + 0.419i)17-s + (0.976 + 0.214i)19-s + (0.856 + 0.515i)20-s + (−0.561 − 0.827i)22-s + (0.161 + 0.986i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.999 + 0.0392i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ 0.999 + 0.0392i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.041878355 + 0.07943158016i\)
\(L(\frac12)\) \(\approx\) \(4.041878355 + 0.07943158016i\)
\(L(1)\) \(\approx\) \(2.249626053 - 0.08020651625i\)
\(L(1)\) \(\approx\) \(2.249626053 - 0.08020651625i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.947 - 0.319i)T \)
5 \( 1 + (0.370 + 0.928i)T \)
7 \( 1 + (0.907 + 0.419i)T \)
11 \( 1 + (-0.267 - 0.963i)T \)
13 \( 1 + (0.468 + 0.883i)T \)
17 \( 1 + (-0.907 + 0.419i)T \)
19 \( 1 + (0.976 + 0.214i)T \)
23 \( 1 + (0.161 + 0.986i)T \)
29 \( 1 + (0.947 + 0.319i)T \)
31 \( 1 + (0.976 - 0.214i)T \)
37 \( 1 + (-0.561 - 0.827i)T \)
41 \( 1 + (0.161 - 0.986i)T \)
43 \( 1 + (0.267 - 0.963i)T \)
47 \( 1 + (0.370 - 0.928i)T \)
53 \( 1 + (-0.647 + 0.762i)T \)
61 \( 1 + (-0.947 + 0.319i)T \)
67 \( 1 + (-0.561 + 0.827i)T \)
71 \( 1 + (0.370 - 0.928i)T \)
73 \( 1 + (-0.994 - 0.108i)T \)
79 \( 1 + (-0.856 - 0.515i)T \)
83 \( 1 + (-0.0541 + 0.998i)T \)
89 \( 1 + (0.947 + 0.319i)T \)
97 \( 1 + (-0.994 + 0.108i)T \)
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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

−27.076277324099843802353566648365, −25.9337351346366740468251785569, −24.803404737106376075682053321505, −24.44642153100828093755966721327, −23.30179846358678117977978375964, −22.52379447446235678985175714046, −21.25319589564568977569485916440, −20.49274390652598382325781597579, −20.06092626594723102835084510106, −17.87686116737003065783336367242, −17.39882826220837857996043893654, −16.120014520492120472086736828560, −15.33587794829899005382569687212, −14.12625025640346143077035220459, −13.30970104275094517199310122835, −12.414257619277409741158688932161, −11.358885264437285839435167547422, −10.12211925994029132628418334747, −8.51331950529000300016559770360, −7.63460464264306582131715790771, −6.308430647377314810533507063, −4.90521336629031936340499762346, −4.55705974987647105350417032648, −2.71875023294109448600892487902, −1.27229756471746695034714914184, 1.56231749133526072479165510921, 2.71280188035853690148391564691, 3.91999419848437627308739922643, 5.34106997268136312500305109020, 6.228433072346743155570415088024, 7.396250163369128582170182297004, 8.937689248109994795798251411241, 10.46732952742699699906086014530, 11.219440721710407655286945581511, 11.99730764651778329964476364139, 13.75360924345207279482884594832, 13.90236422245836081736055175791, 15.1725009966147288799528891151, 15.95878372870756345696481077581, 17.52501307925356800952071729255, 18.60063681938142404870239201596, 19.37869680879483893121899932060, 20.79276913014616820271760125420, 21.54606838718849615826684478542, 22.10152224006672102559858964889, 23.30836395586583418903288078429, 24.15424003161699526486370979256, 24.97947390482582644687702385315, 26.17091033557887057148601216648, 27.11586175573876261819811114623

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