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Properties

Label 1-89-89.6-r1-0-0
Degree $1$
Conductor $89$
Sign $-0.998 + 0.0573i$
Analytic cond. $9.56437$
Root an. cond. $9.56437$
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  + (0.841 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.998 + 0.0573i$
Analytic conductor: \(9.56437\)
Root analytic conductor: \(9.56437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (1:\ ),\ -0.998 + 0.0573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05186143191 + 1.807388460i\)
\(L(\frac12)\) \(\approx\) \(0.05186143191 + 1.807388460i\)
\(L(1)\) \(\approx\) \(0.9488997203 + 0.9979451985i\)
\(L(1)\) \(\approx\) \(0.9488997203 + 0.9979451985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (0.841 + 0.540i)T \)
3 \( 1 + (0.349 + 0.936i)T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.599 - 0.800i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (-0.936 + 0.349i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (0.0713 - 0.997i)T \)
23 \( 1 + (0.997 + 0.0713i)T \)
29 \( 1 + (-0.800 + 0.599i)T \)
31 \( 1 + (0.997 - 0.0713i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.936 + 0.349i)T \)
43 \( 1 + (-0.800 - 0.599i)T \)
47 \( 1 + (-0.909 + 0.415i)T \)
53 \( 1 + (0.909 + 0.415i)T \)
59 \( 1 + (-0.349 + 0.936i)T \)
61 \( 1 + (-0.479 + 0.877i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (0.989 - 0.142i)T \)
73 \( 1 + (0.654 - 0.755i)T \)
79 \( 1 + (0.755 + 0.654i)T \)
83 \( 1 + (0.212 - 0.977i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

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

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