Properties

Label 1-177-177.170-r0-0-0
Degree $1$
Conductor $177$
Sign $0.662 - 0.749i$
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  + (−0.370 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (170, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.662 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3861508902 - 0.1741081483i\)
\(L(\frac12)\) \(\approx\) \(0.3861508902 - 0.1741081483i\)
\(L(1)\) \(\approx\) \(0.5706092964 + 0.1160921514i\)
\(L(1)\) \(\approx\) \(0.5706092964 + 0.1160921514i\)

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

−27.76291377977594593120881752212, −26.85521967116756939693462599902, −25.689656990702039249713583145590, −24.86755382154709204661368867657, −23.35398887757918546113949618522, −22.67465424662985622878257637482, −21.587062631504620416820312605730, −20.60209806113955354246239358208, −19.64123668010278634233174059369, −19.18022360952312635262693975167, −17.87689381105787392974195864835, −16.88275354319727245928708368230, −15.9400550032522277975779505308, −14.68067733666093741122874493962, −13.14599821120227942025880880170, −12.24395603923896740799477149293, −11.930854877570758562087927973596, −10.12274498695456134888369045664, −9.576257485259325838491354012438, −8.32279072370887279149578797931, −7.38217012868244602509424515589, −5.45478730755421469418008938057, −4.224945519651233376211515748759, −3.09595135273686277556357077024, −1.589785146136245398613719005751, 0.3966383830828317670526767664, 2.975629541259456016785068179780, 4.24819699377371656855526981990, 5.74663661409556386326090500316, 6.924731020498757658080183382125, 7.51648493176658458859152571898, 8.86905214689726496222108128217, 9.99746771737473514119485189979, 10.91803882534873359967464101394, 12.36738553579876338647814623316, 13.81977822745318894583133920692, 14.433060298896427242005761265551, 15.69336023157836749741872382603, 16.3370584084497160357028739312, 17.323697105799669826557792367358, 18.65266259412843534430804724798, 19.1746757023590787262306298777, 20.0992188430013961733248341820, 22.029339099803706967855810342582, 22.52425034550744546532899077354, 23.638860266239069964711165812440, 24.27441021720311851078479984563, 25.56949367231774369902559105298, 26.403309511668170696698544004992, 26.914305450695328787848755764414

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