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Properties

Label 1-89-89.71-r0-0-0
Degree $1$
Conductor $89$
Sign $-0.0526 + 0.998i$
Analytic cond. $0.413314$
Root an. cond. $0.413314$
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  + (−0.142 + 0.989i)2-s + (−0.281 − 0.959i)3-s + (−0.959 − 0.281i)4-s + (−0.415 + 0.909i)5-s + (0.989 − 0.142i)6-s + (0.909 + 0.415i)7-s + (0.415 − 0.909i)8-s + (−0.841 + 0.540i)9-s + (−0.841 − 0.540i)10-s + (0.415 + 0.909i)11-s + i·12-s + (0.281 + 0.959i)13-s + (−0.540 + 0.841i)14-s + (0.989 + 0.142i)15-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)17-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)2-s + (−0.281 − 0.959i)3-s + (−0.959 − 0.281i)4-s + (−0.415 + 0.909i)5-s + (0.989 − 0.142i)6-s + (0.909 + 0.415i)7-s + (0.415 − 0.909i)8-s + (−0.841 + 0.540i)9-s + (−0.841 − 0.540i)10-s + (0.415 + 0.909i)11-s + i·12-s + (0.281 + 0.959i)13-s + (−0.540 + 0.841i)14-s + (0.989 + 0.142i)15-s + (0.841 + 0.540i)16-s + (0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0526 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0526 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $-0.0526 + 0.998i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ -0.0526 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5091910925 + 0.5367682566i\)
\(L(\frac12)\) \(\approx\) \(0.5091910925 + 0.5367682566i\)
\(L(1)\) \(\approx\) \(0.7200886756 + 0.3822748252i\)
\(L(1)\) \(\approx\) \(0.7200886756 + 0.3822748252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad89 \( 1 \)
good2 \( 1 + (-0.142 + 0.989i)T \)
3 \( 1 + (-0.281 - 0.959i)T \)
5 \( 1 + (-0.415 + 0.909i)T \)
7 \( 1 + (0.909 + 0.415i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (-0.540 - 0.841i)T \)
23 \( 1 + (0.540 + 0.841i)T \)
29 \( 1 + (-0.909 - 0.415i)T \)
31 \( 1 + (0.540 - 0.841i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.281 - 0.959i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.281 + 0.959i)T \)
61 \( 1 + (-0.755 - 0.654i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (-0.841 - 0.540i)T \)
83 \( 1 + (0.989 - 0.142i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−30.10704962091435860151537924539, −29.08510831939896434751418074663, −27.95889838139846656138259903795, −27.31288443742184766617170157821, −26.7869367390454501502291042323, −24.98429897932421554324954885759, −23.543464395375491679593376641939, −22.656388013337234577699197183225, −21.38695487543621732792735165376, −20.63238389864773619834122390144, −20.01209812112895096452763610797, −18.46347662275327746670746649917, −17.136721828885173760633772499656, −16.45962053433254189612514761521, −14.86993430330217852643727171264, −13.630630730736198275360489653934, −12.17087430674064709244863046425, −11.251182531560792556895433163737, −10.33269120903745439252265969087, −8.91561478956014316883880491213, −8.15573362007295431537206039602, −5.40745397996761285550395862701, −4.46735835379596856840948530473, −3.31725364530825656387875220739, −0.96953527722124722383002324959, 1.896458691234406766849769227362, 4.262012000810423030539523183655, 5.85941731525629692957690586982, 6.95716262026204558265967059100, 7.75674333347310237620895394388, 9.027060574241981783981131055432, 10.8893675214762232730606700697, 12.00868010975480502120792550476, 13.455953880005420076905374688357, 14.60635821618650170412173769346, 15.2756078733742466865222673887, 17.03652696919004133171865533232, 17.75734178916576974002827833254, 18.73971985256432308326123971647, 19.49533866224676921149850768412, 21.58637952986511605290806820562, 22.739101055124861316289042788409, 23.60045641902588611124636080301, 24.33601836479218366175661930541, 25.52673973180029606690604791494, 26.274835027942461134948691858896, 27.70693273169837326198701979561, 28.346799136254785754275778177713, 30.14369905711646592700784875526, 30.78989484810711418183521312295

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