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Properties

Label 1-177-177.158-r0-0-0
Degree $1$
Conductor $177$
Sign $-0.188 - 0.982i$
Analytic cond. $0.821984$
Root an. cond. $0.821984$
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  + (−0.561 − 0.827i)2-s + (−0.370 + 0.928i)4-s + (−0.907 + 0.419i)5-s + (0.267 − 0.963i)7-s + (0.976 − 0.214i)8-s + (0.856 + 0.515i)10-s + (−0.725 + 0.687i)11-s + (0.994 − 0.108i)13-s + (−0.947 + 0.319i)14-s + (−0.725 − 0.687i)16-s + (−0.267 − 0.963i)17-s + (0.796 − 0.605i)19-s + (−0.0541 − 0.998i)20-s + (0.976 + 0.214i)22-s + (0.468 − 0.883i)23-s + ⋯
L(s)  = 1  + (−0.561 − 0.827i)2-s + (−0.370 + 0.928i)4-s + (−0.907 + 0.419i)5-s + (0.267 − 0.963i)7-s + (0.976 − 0.214i)8-s + (0.856 + 0.515i)10-s + (−0.725 + 0.687i)11-s + (0.994 − 0.108i)13-s + (−0.947 + 0.319i)14-s + (−0.725 − 0.687i)16-s + (−0.267 − 0.963i)17-s + (0.796 − 0.605i)19-s + (−0.0541 − 0.998i)20-s + (0.976 + 0.214i)22-s + (0.468 − 0.883i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.188 - 0.982i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ -0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4181786448 - 0.5058684335i\)
\(L(\frac12)\) \(\approx\) \(0.4181786448 - 0.5058684335i\)
\(L(1)\) \(\approx\) \(0.6149340334 - 0.3145897936i\)
\(L(1)\) \(\approx\) \(0.6149340334 - 0.3145897936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.561 - 0.827i)T \)
5 \( 1 + (-0.907 + 0.419i)T \)
7 \( 1 + (0.267 - 0.963i)T \)
11 \( 1 + (-0.725 + 0.687i)T \)
13 \( 1 + (0.994 - 0.108i)T \)
17 \( 1 + (-0.267 - 0.963i)T \)
19 \( 1 + (0.796 - 0.605i)T \)
23 \( 1 + (0.468 - 0.883i)T \)
29 \( 1 + (0.561 - 0.827i)T \)
31 \( 1 + (-0.796 - 0.605i)T \)
37 \( 1 + (-0.976 - 0.214i)T \)
41 \( 1 + (-0.468 - 0.883i)T \)
43 \( 1 + (0.725 + 0.687i)T \)
47 \( 1 + (0.907 + 0.419i)T \)
53 \( 1 + (0.856 - 0.515i)T \)
61 \( 1 + (0.561 + 0.827i)T \)
67 \( 1 + (-0.976 + 0.214i)T \)
71 \( 1 + (-0.907 - 0.419i)T \)
73 \( 1 + (0.947 - 0.319i)T \)
79 \( 1 + (0.0541 + 0.998i)T \)
83 \( 1 + (-0.161 - 0.986i)T \)
89 \( 1 + (-0.561 + 0.827i)T \)
97 \( 1 + (0.947 + 0.319i)T \)
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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

−27.566681753733486558884077334931, −26.76388849038057654221113985475, −25.71805389403321465056360945279, −24.813030388717358341367296067598, −23.8468556961746069306603949669, −23.36664252124907727927498314196, −22.00537744656372539487316767139, −20.80888135360620580673977152302, −19.60343643107921539988600999502, −18.736391817421841036716404332383, −18.04431211575149677378370732848, −16.69443621180962787686999299765, −15.78784486020075851856434665124, −15.32308834112975552727243401710, −14.02992830153225676676847996264, −12.80480582343105246994457969760, −11.49946289105963402537435355781, −10.529995448830658684553310629468, −8.871065080389657067826933451680, −8.46268720813449535152458845450, −7.3792561251647679774175095234, −5.90809252389623359946508663407, −5.08765271247384241075899090325, −3.51743377147687692714558617447, −1.41325019966318255948761875029, 0.72592890205395937202136102639, 2.56388795932863114830779413777, 3.76660520148432615493582863352, 4.752056398886626041129154803751, 7.05847272102077912993077437064, 7.685862556153979784984228088457, 8.86471024921540767311732063570, 10.25215841535655687166447142846, 10.97261588528265597863720145989, 11.829353754407734662261365829543, 13.06996092029915096794401824306, 14.00279797316523694306872850322, 15.53556492050101906509345379948, 16.392351597930052045592136649758, 17.71535122173269785848445680289, 18.38326230628329752748537105142, 19.424493020402788096839340318296, 20.46273047758559762275388984297, 20.79951713141817252624070372545, 22.514487448836306894192237697153, 22.96842839233195764282884294958, 24.091070310623758657378073139369, 25.65493394341282249658484646550, 26.47399953762975238499533000933, 27.10304883020874404166703902105

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