Properties

Degree 1
Conductor $ 3 \cdot 29 $
Sign $0.801 - 0.598i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(87\)    =    \(3 \cdot 29\)
\( \varepsilon \)  =  $0.801 - 0.598i$
motivic weight  =  \(0\)
character  :  $\chi_{87} (8, \cdot )$
Sato-Tate  :  $\mu(28)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 87,\ (0:\ ),\ 0.801 - 0.598i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7324454015 - 0.2434028704i$
$L(\frac12,\chi)$  $\approx$  $0.7324454015 - 0.2434028704i$
$L(\chi,1)$  $\approx$  0.7956699705 - 0.2039708339i
$L(1,\chi)$  $\approx$  0.7956699705 - 0.2039708339i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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