Properties

Degree 1
Conductor 89
Sign $-0.998 - 0.0573i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.841 − 0.540i)2-s + (0.349 − 0.936i)3-s + (0.415 − 0.909i)4-s + (−0.989 + 0.142i)5-s + (−0.212 − 0.977i)6-s + (−0.599 + 0.800i)7-s + (−0.142 − 0.989i)8-s + (−0.755 − 0.654i)9-s + (−0.755 + 0.654i)10-s + (0.142 − 0.989i)11-s + (−0.707 − 0.707i)12-s + (−0.936 − 0.349i)13-s + (−0.0713 + 0.997i)14-s + (−0.212 + 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + ⋯
L(s,χ)  = 1  + (0.841 − 0.540i)2-s + (0.349 − 0.936i)3-s + (0.415 − 0.909i)4-s + (−0.989 + 0.142i)5-s + (−0.212 − 0.977i)6-s + (−0.599 + 0.800i)7-s + (−0.142 − 0.989i)8-s + (−0.755 − 0.654i)9-s + (−0.755 + 0.654i)10-s + (0.142 − 0.989i)11-s + (−0.707 − 0.707i)12-s + (−0.936 − 0.349i)13-s + (−0.0713 + 0.997i)14-s + (−0.212 + 0.977i)15-s + (−0.654 − 0.755i)16-s + (0.540 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.998 - 0.0573i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.998 - 0.0573i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(89\)
\( \varepsilon \)  =  $-0.998 - 0.0573i$
motivic weight  =  \(0\)
character  :  $\chi_{89} (15, \cdot )$
Sato-Tate  :  $\mu(88)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 89,\ (1:\ ),\ -0.998 - 0.0573i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.05186143191 - 1.807388460i$
$L(\frac12,\chi)$  $\approx$  $0.05186143191 - 1.807388460i$
$L(\chi,1)$  $\approx$  0.9488997203 - 0.9979451985i
$L(1,\chi)$  $\approx$  0.9488997203 - 0.9979451985i

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

−31.05373724747791017152609504592, −30.17057279710668215537125771443, −28.61875880634386063694493955591, −27.33746242224381907202269499831, −26.38841230962544226328943327614, −25.697615147318771261971261729933, −24.27560528076460582439095262991, −23.16683110109142187461154599047, −22.50706970333327414257689842984, −21.32594220119192370450071231731, −20.14389703190167565404435171037, −19.544504112648201855585416963355, −17.12287047641995159897146677251, −16.50394719445600614781855485586, −15.261675824077051460917149802185, −14.76580499478378290451470478797, −13.30991190021217524177701901804, −12.126935078664571199636306846462, −10.83580760824067808982747067858, −9.372607072441074635196709294188, −7.865933480542954854866727317804, −6.88128727273119057722544586067, −4.89793199088561317984177866556, −4.13198425906419695738926590754, −2.97915868206805612735597012671, 0.596379096472079233404395400438, 2.617076115110116722528777983501, 3.48500709626464346067590321414, 5.43128033677049260312948831026, 6.67921640700083234196667011504, 7.98541513840074682137181961234, 9.52094823764089138971332917416, 11.33368935882207133364621153725, 12.129836103257730787421034685018, 12.95744842874564835113619194137, 14.285627711928025701539373445784, 15.161013888794608833314285261139, 16.43283711196631740950201391450, 18.56455886983291878855849093357, 19.10649888361868300197540303414, 19.901869649264132887956198149645, 21.18398777412621106532589126286, 22.613545990580230507106947617329, 23.15311186096471532381864438573, 24.566539769524684865540185857170, 24.89591349147154960119148546645, 26.65900406458141567882266765927, 27.89154863823848296507947993387, 29.18526482620506309313122586752, 29.747086166383477759932312802013

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