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Properties

Label 1-87-87.8-r0-0-0
Degree $1$
Conductor $87$
Sign $0.801 - 0.598i$
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  + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.222 + 0.974i)7-s + (0.433 − 0.900i)8-s + (−0.974 + 0.222i)10-s + (0.433 + 0.900i)11-s + (0.900 − 0.433i)13-s + (0.781 − 0.623i)14-s + (−0.900 + 0.433i)16-s − i·17-s + (0.974 − 0.222i)19-s + (0.900 + 0.433i)20-s + (0.222 − 0.974i)22-s + (−0.623 − 0.781i)23-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (0.623 − 0.781i)5-s + (−0.222 + 0.974i)7-s + (0.433 − 0.900i)8-s + (−0.974 + 0.222i)10-s + (0.433 + 0.900i)11-s + (0.900 − 0.433i)13-s + (0.781 − 0.623i)14-s + (−0.900 + 0.433i)16-s − i·17-s + (0.974 − 0.222i)19-s + (0.900 + 0.433i)20-s + (0.222 − 0.974i)22-s + (−0.623 − 0.781i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(87\)    =    \(3 \cdot 29\)
Sign: $0.801 - 0.598i$
Analytic conductor: \(0.404026\)
Root analytic conductor: \(0.404026\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{87} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 87,\ (0:\ ),\ 0.801 - 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7324454015 - 0.2434028704i\)
\(L(\frac12)\) \(\approx\) \(0.7324454015 - 0.2434028704i\)
\(L(1)\) \(\approx\) \(0.7956699705 - 0.2039708339i\)
\(L(1)\) \(\approx\) \(0.7956699705 - 0.2039708339i\)

Euler product

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

−30.43912646291882171514041313462, −29.557045571120877661663010463162, −28.63214630329896452194403339258, −27.307388277828268689608019331062, −26.316304227644498423531847836143, −25.85024353198713342681157033669, −24.50757196465938185079285073445, −23.51209751333722923239970634873, −22.41084090535597056036927912600, −21.01767358930123273433866610951, −19.64392271783492798050316374743, −18.75787227013895208953704755709, −17.64813499533304751090440674281, −16.751619681646192224253441358900, −15.65215124626032153856207515400, −14.16648566003477027832834838578, −13.66094321477380874483153897112, −11.29674756251294824426111873356, −10.41917369339735598077076887906, −9.34416335405109925188592026735, −7.914939787638110490423574320038, −6.656168846153458704363209353469, −5.816875453960933637306866009132, −3.62781179531452867102808494481, −1.50384481932710175681121136423, 1.467691017332201660728238547527, 2.91271147710476679793753861736, 4.825779780917565134051361772949, 6.46508966418033570614080274415, 8.19106933265219049548930965104, 9.21410967639845365694516778571, 10.00995323366046871835422235839, 11.69825167987481262051310997888, 12.49734536178408400641680242972, 13.663071489109686721123431459053, 15.58231712386484405829879897359, 16.52311708517069584329764671473, 17.84538207091775025395153026440, 18.40583654685562014991359114424, 19.97869190500119419871986043998, 20.63927241321878262891287683460, 21.74679877864408156350525744107, 22.75619153358067403279867331895, 24.78584503253480084752697156986, 25.18326401740441340039011573232, 26.33435802995298369508360422383, 27.84835853126891929212685712965, 28.28392739452329815394917741705, 29.19366371287128468468216442227, 30.39899995267208755549671318899

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