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Properties

Label 1-177-177.137-r1-0-0
Degree $1$
Conductor $177$
Sign $0.843 - 0.536i$
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  + (0.561 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.843 - 0.536i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ 0.843 - 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254960635 - 0.3654972640i\)
\(L(\frac12)\) \(\approx\) \(1.254960635 - 0.3654972640i\)
\(L(1)\) \(\approx\) \(1.037439993 + 0.2784564954i\)
\(L(1)\) \(\approx\) \(1.037439993 + 0.2784564954i\)

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

−27.59889960286054587136574007489, −26.63390013684299763791970031885, −24.84612430729919186543259942686, −24.36851010576270370866362661933, −23.217149563478266620474504310661, −22.32825041834425172556206367099, −21.568457941643798708635115554625, −20.310205000690509350235654089046, −19.73771402940527993003310034297, −18.78184375484477400768248230827, −17.72118488197077265625002973848, −16.27891124781167433114192975408, −14.964927439149123431540195639013, −14.65264045280173849650369063513, −12.93925349708801241112569859772, −12.1037286939350503582411312370, −11.65849141205892952430626520554, −10.19323753792737253232994851419, −9.120971834857352222050692981398, −8.002291293214179887270913582145, −6.35190787473645854418975181932, −4.99242013685604982349205474136, −4.17554590600379457577870788767, −2.77341404377437032217780856943, −1.40777399996985105187318655594, 0.41131290805254540954788922732, 3.01091696472301272706075547709, 4.05584147662560600899418483284, 5.040787217490493460690189753080, 6.69608757406165767428552138320, 7.347510052946455513539842670615, 8.328249837347785183850357020581, 9.75266247520049141992677086935, 11.41409222083108910959882161069, 11.95268724686267333391817394823, 13.59415246064680141231839534828, 14.151147872201885915929770509840, 15.2486386389303897772669765757, 16.15528419308111297038059112356, 17.07123791488612559971890562605, 18.03493211374000334285479452507, 19.42057047090856057495726632800, 20.24330875794329027503124878477, 21.64627971267484890488088936419, 22.47099811937635169691268695183, 23.309450102910602191163982621071, 24.15362986115204028494277329386, 24.871432529459065349430710437295, 26.304560724504162727800981240864, 26.90989871313501455017450518293

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