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Properties

Label 1-177-177.128-r0-0-0
Degree $1$
Conductor $177$
Sign $0.0900 + 0.995i$
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  + (−0.725 − 0.687i)2-s + (0.0541 + 0.998i)4-s + (0.161 + 0.986i)5-s + (0.468 + 0.883i)7-s + (0.647 − 0.762i)8-s + (0.561 − 0.827i)10-s + (−0.994 − 0.108i)11-s + (−0.907 + 0.419i)13-s + (0.267 − 0.963i)14-s + (−0.994 + 0.108i)16-s + (−0.468 + 0.883i)17-s + (−0.856 − 0.515i)19-s + (−0.976 + 0.214i)20-s + (0.647 + 0.762i)22-s + (−0.370 + 0.928i)23-s + ⋯
L(s)  = 1  + (−0.725 − 0.687i)2-s + (0.0541 + 0.998i)4-s + (0.161 + 0.986i)5-s + (0.468 + 0.883i)7-s + (0.647 − 0.762i)8-s + (0.561 − 0.827i)10-s + (−0.994 − 0.108i)11-s + (−0.907 + 0.419i)13-s + (0.267 − 0.963i)14-s + (−0.994 + 0.108i)16-s + (−0.468 + 0.883i)17-s + (−0.856 − 0.515i)19-s + (−0.976 + 0.214i)20-s + (0.647 + 0.762i)22-s + (−0.370 + 0.928i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0900 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0900 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.0900 + 0.995i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.0900 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4424890225 + 0.4042664035i\)
\(L(\frac12)\) \(\approx\) \(0.4424890225 + 0.4042664035i\)
\(L(1)\) \(\approx\) \(0.6602676099 + 0.1150557210i\)
\(L(1)\) \(\approx\) \(0.6602676099 + 0.1150557210i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.725 - 0.687i)T \)
5 \( 1 + (0.161 + 0.986i)T \)
7 \( 1 + (0.468 + 0.883i)T \)
11 \( 1 + (-0.994 - 0.108i)T \)
13 \( 1 + (-0.907 + 0.419i)T \)
17 \( 1 + (-0.468 + 0.883i)T \)
19 \( 1 + (-0.856 - 0.515i)T \)
23 \( 1 + (-0.370 + 0.928i)T \)
29 \( 1 + (0.725 - 0.687i)T \)
31 \( 1 + (0.856 - 0.515i)T \)
37 \( 1 + (-0.647 - 0.762i)T \)
41 \( 1 + (0.370 + 0.928i)T \)
43 \( 1 + (0.994 - 0.108i)T \)
47 \( 1 + (-0.161 + 0.986i)T \)
53 \( 1 + (0.561 + 0.827i)T \)
61 \( 1 + (0.725 + 0.687i)T \)
67 \( 1 + (-0.647 + 0.762i)T \)
71 \( 1 + (0.161 - 0.986i)T \)
73 \( 1 + (-0.267 + 0.963i)T \)
79 \( 1 + (0.976 - 0.214i)T \)
83 \( 1 + (0.796 - 0.605i)T \)
89 \( 1 + (-0.725 + 0.687i)T \)
97 \( 1 + (-0.267 - 0.963i)T \)
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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

−27.14677988703261266393162387644, −26.29842402073313047656830209844, −25.19372304422040461073259000004, −24.3643787292745933330642377267, −23.69793892319229836198779281383, −22.71227237162385681886718327961, −20.99409016560830306042073547336, −20.32634890321100451751811873660, −19.41506194610174603974564182719, −18.064791376861904560529421507555, −17.355359617580183323560163320705, −16.49551608432068681531197955889, −15.63993603811110785831432064083, −14.40431500721909071210914882446, −13.46065132430956071074602135439, −12.225432827268591832231761161478, −10.64301916665018409988921131478, −9.97386014919092208607178276885, −8.62417628265474064728005354134, −7.87807821918139234213282985118, −6.77635571298367093145378347100, −5.23974504961773009491314984538, −4.57691561896054360363167484672, −2.17267688548215719986566484120, −0.58396569070809752316348865267, 2.10744049349503933535693607692, 2.7351062262919998180468114537, 4.38048159373769942805209434068, 6.04778148028645691986867154587, 7.39889720709378121835781259614, 8.35305085712192752013027542776, 9.562379995842849383244315628989, 10.56273292574565603227946621686, 11.392031687669186243961754660484, 12.415438778184615717848756144473, 13.59137526168742502913829542901, 14.95290886471633603198293893504, 15.79140215869181815581740504835, 17.418499654175643212064563353941, 17.86275570937979041654245270735, 19.042804503084710960572397755359, 19.44914376875205830995881048124, 21.14559335971058525265785321876, 21.54045839493527399186485963777, 22.444144582089253942288810406197, 23.82698372125548646744628039866, 25.0334027871045128826494250863, 26.07836220953793687289223797578, 26.57992963804120942563572862174, 27.70208981891333110040721488733

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