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Properties

Label 1-177-177.125-r1-0-0
Degree $1$
Conductor $177$
Sign $0.929 - 0.368i$
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  + (0.370 − 0.928i)2-s + (−0.725 − 0.687i)4-s + (−0.647 + 0.762i)5-s + (−0.856 − 0.515i)7-s + (−0.907 + 0.419i)8-s + (0.468 + 0.883i)10-s + (−0.0541 + 0.998i)11-s + (0.976 − 0.214i)13-s + (−0.796 + 0.605i)14-s + (0.0541 + 0.998i)16-s + (0.856 − 0.515i)17-s + (0.267 − 0.963i)19-s + (0.994 − 0.108i)20-s + (0.907 + 0.419i)22-s + (0.561 + 0.827i)23-s + ⋯
L(s)  = 1  + (0.370 − 0.928i)2-s + (−0.725 − 0.687i)4-s + (−0.647 + 0.762i)5-s + (−0.856 − 0.515i)7-s + (−0.907 + 0.419i)8-s + (0.468 + 0.883i)10-s + (−0.0541 + 0.998i)11-s + (0.976 − 0.214i)13-s + (−0.796 + 0.605i)14-s + (0.0541 + 0.998i)16-s + (0.856 − 0.515i)17-s + (0.267 − 0.963i)19-s + (0.994 − 0.108i)20-s + (0.907 + 0.419i)22-s + (0.561 + 0.827i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.463447587 - 0.2790803095i\)
\(L(\frac12)\) \(\approx\) \(1.463447587 - 0.2790803095i\)
\(L(1)\) \(\approx\) \(0.9827598140 - 0.3365968645i\)
\(L(1)\) \(\approx\) \(0.9827598140 - 0.3365968645i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.370 - 0.928i)T \)
5 \( 1 + (-0.647 + 0.762i)T \)
7 \( 1 + (-0.856 - 0.515i)T \)
11 \( 1 + (-0.0541 + 0.998i)T \)
13 \( 1 + (0.976 - 0.214i)T \)
17 \( 1 + (0.856 - 0.515i)T \)
19 \( 1 + (0.267 - 0.963i)T \)
23 \( 1 + (0.561 + 0.827i)T \)
29 \( 1 + (0.370 + 0.928i)T \)
31 \( 1 + (0.267 + 0.963i)T \)
37 \( 1 + (0.907 + 0.419i)T \)
41 \( 1 + (0.561 - 0.827i)T \)
43 \( 1 + (0.0541 + 0.998i)T \)
47 \( 1 + (-0.647 - 0.762i)T \)
53 \( 1 + (-0.468 + 0.883i)T \)
61 \( 1 + (-0.370 + 0.928i)T \)
67 \( 1 + (0.907 - 0.419i)T \)
71 \( 1 + (-0.647 - 0.762i)T \)
73 \( 1 + (0.796 - 0.605i)T \)
79 \( 1 + (-0.994 + 0.108i)T \)
83 \( 1 + (0.947 - 0.319i)T \)
89 \( 1 + (0.370 + 0.928i)T \)
97 \( 1 + (0.796 + 0.605i)T \)
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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

−27.04750529712211871416581106402, −26.13138309055215645729944834275, −25.11845312503162597422685158363, −24.41900237656511097774867628201, −23.36461622893026534361991638023, −22.77405646795804048787973004382, −21.518133411925223374799956537, −20.71352488486095142214849624274, −19.114853764901980950450057760745, −18.59468552083814786141458458420, −16.93656594129202704816042946658, −16.28920182674772575105296539589, −15.65922608079343967843032653080, −14.465571632570132842081697838699, −13.25375681492294039240567139842, −12.55600307156786241977876825159, −11.472500118561167114391595118, −9.663125980889763625863666166302, −8.564780264161197036271969816660, −7.92647328522419045131765667381, −6.312117764302232337143933733595, −5.63144028089371730661836827830, −4.14002552505637198685872847081, −3.227329282995571499879600609528, −0.6415244717592786958545852367, 0.994848054418276180403514524382, 2.86979623337602628061445326104, 3.59576705705175433258789226839, 4.868906331380370456764788141554, 6.42443937049761713702822938269, 7.515358905888690849070269425932, 9.16340661716418640965707023655, 10.18669390258851089273626788533, 11.01807529340940525724869157111, 12.065226806328545246419577510773, 13.07619775740619272446324780366, 14.03389747472137150248793118449, 15.13916143423330080355204660190, 16.05407587922903431759153383221, 17.717983665617289246213628910180, 18.54231663671638434741399805287, 19.57899365672471730578774967856, 20.13522208182262673156645021173, 21.30727555117942272842342314257, 22.438982220628143582108410734998, 23.14832935488581909771988671302, 23.57288883041840957540217136782, 25.4065107888211945896375701461, 26.25272965188640222133085792021, 27.31973863862940321374227396715

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