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Properties

Label 1-87-87.68-r0-0-0
Degree $1$
Conductor $87$
Sign $0.995 - 0.0997i$
Analytic cond. $0.404026$
Root an. cond. $0.404026$
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  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)5-s + (0.623 − 0.781i)7-s + (0.974 − 0.222i)8-s + (0.781 − 0.623i)10-s + (0.974 + 0.222i)11-s + (0.222 − 0.974i)13-s + (0.433 + 0.900i)14-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.781 + 0.623i)19-s + (0.222 + 0.974i)20-s + (−0.623 + 0.781i)22-s + (0.900 − 0.433i)23-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (−0.623 − 0.781i)4-s + (−0.900 − 0.433i)5-s + (0.623 − 0.781i)7-s + (0.974 − 0.222i)8-s + (0.781 − 0.623i)10-s + (0.974 + 0.222i)11-s + (0.222 − 0.974i)13-s + (0.433 + 0.900i)14-s + (−0.222 + 0.974i)16-s − i·17-s + (−0.781 + 0.623i)19-s + (0.222 + 0.974i)20-s + (−0.623 + 0.781i)22-s + (0.900 − 0.433i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.995 - 0.0997i$
Analytic conductor: \(0.404026\)
Root analytic conductor: \(0.404026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (0:\ ),\ 0.995 - 0.0997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6955635165 - 0.03476680458i\)
\(L(\frac12)\) \(\approx\) \(0.6955635165 - 0.03476680458i\)
\(L(1)\) \(\approx\) \(0.7644587223 + 0.08402910098i\)
\(L(1)\) \(\approx\) \(0.7644587223 + 0.08402910098i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.433 + 0.900i)T \)
5 \( 1 + (-0.900 - 0.433i)T \)
7 \( 1 + (0.623 - 0.781i)T \)
11 \( 1 + (0.974 + 0.222i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.781 + 0.623i)T \)
23 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.433 - 0.900i)T \)
37 \( 1 + (-0.974 + 0.222i)T \)
41 \( 1 + iT \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (0.781 + 0.623i)T \)
67 \( 1 + (0.222 + 0.974i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.433 + 0.900i)T \)
79 \( 1 + (0.974 - 0.222i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.433 + 0.900i)T \)
97 \( 1 + (0.781 - 0.623i)T \)
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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

−30.73514942026164304387000112153, −29.69723055814498570406504035614, −28.30279329174246934038692885713, −27.68149679130188100274771358881, −26.745531127389026743828022801549, −25.688847478806361998204580185076, −24.20963296890795461858664129229, −23.01061883213932677207471939630, −21.822304962139476013384465149564, −21.12881544146923272320641322746, −19.47596792079739411765948164293, −19.172370242360127086372394300185, −17.8899488047536465023697588316, −16.763947815572755154798282819542, −15.261273694926370803788994662109, −14.12462751628266548940937966380, −12.48102162141166837398924929792, −11.56555671697906283591914794865, −10.828795732215320060949493574060, −9.07675273154434479157061142269, −8.31223790796670270048965935818, −6.76140068455620403969994644849, −4.59807294545175725664841135302, −3.40409675488159037735244571356, −1.76278656609422774576018981287, 1.01007176919940394945893374550, 3.97417692785062555574987915012, 5.06203221410069970624827036409, 6.79279642265836277308239976997, 7.83810701058061526299049304319, 8.783825146700662472374161644639, 10.27662986874787205675652349200, 11.55710404910772801422816385165, 13.13389693814877893199516297356, 14.46756638739633890457510713476, 15.342124879814550961743754827489, 16.602420297174763054642563480949, 17.30641196028535679243521666918, 18.623647278255749721381931918233, 19.78781393825064781464993918608, 20.64220637410198339610649674529, 22.699869522486939373390494122103, 23.23811717470581743503890710940, 24.44693911864456173030348331077, 25.12592313341964556323220403189, 26.604309694064484255020775872867, 27.45399483323912077472799238136, 27.90048929405822770661229910655, 29.5634956743013136926591278557, 30.78969010453245905848096022475

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