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Properties

Label 1-89-89.81-r0-0-0
Degree $1$
Conductor $89$
Sign $0.763 - 0.645i$
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 + (0.959 + 0.281i)3-s + (−0.959 + 0.281i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (−0.415 − 0.909i)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 − 12-s + (0.959 + 0.281i)13-s + (−0.841 + 0.540i)14-s + (0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.142 + 0.989i)17-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (0.959 + 0.281i)3-s + (−0.959 + 0.281i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (−0.415 − 0.909i)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 − 12-s + (0.959 + 0.281i)13-s + (−0.841 + 0.540i)14-s + (0.142 + 0.989i)15-s + (0.841 − 0.540i)16-s + (−0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(89\)
Sign: $0.763 - 0.645i$
Analytic conductor: \(0.413314\)
Root analytic conductor: \(0.413314\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{89} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 89,\ (0:\ ),\ 0.763 - 0.645i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109039771 - 0.4061931986i\)
\(L(\frac12)\) \(\approx\) \(1.109039771 - 0.4061931986i\)
\(L(1)\) \(\approx\) \(1.154433819 - 0.3493161278i\)
\(L(1)\) \(\approx\) \(1.154433819 - 0.3493161278i\)

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 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (0.415 - 0.909i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 - T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (-0.959 + 0.281i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (0.654 + 0.755i)T \)
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 \( 1 + (0.142 - 0.989i)T \)
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

−31.21244048272459775886250274474, −29.642449360489911198106063486631, −28.159022461108187125525679230301, −27.52348573353662722950272033350, −25.81351577844993301228514621478, −25.43271623994524622776552819012, −24.68246078559688388804256844632, −23.602036798198056050111451812571, −22.265548500036404910938887124751, −20.94681532975844475927155632182, −19.85406010515514324103411030363, −18.58092926539742864551194619676, −17.76790288457884433437199717287, −16.28447851502734777652390567924, −15.44279777619410648012831729987, −14.32235233732114043936658769763, −13.18579143747910419000853237355, −12.40758094910834175134578243460, −9.756818987992662207564639653022, −9.04695724536481340426495294709, −8.146466207633319443480260895279, −6.70668243861503145882512944713, −5.425231581226617798435534294254, −3.89567328533953235998705554238, −1.78082822615183750474694528149, 1.84606870866405506945428976612, 3.32183198015829469526231128559, 4.05319956005539095245042261800, 6.40566416149925412092127830612, 8.081197492591433131460607243723, 9.22443381723153531639094622931, 10.408588886602122437363854472201, 11.03967761268516048604298852197, 13.05118637903412246697868405299, 13.78943319744076698540339230166, 14.651845284537045492843427545801, 16.36630522304764605530244869436, 17.746734352165854850697269856283, 19.06584900502818850049117487170, 19.51943674103121995308192942229, 20.82878063959399530862924775928, 21.62736188797390228691945167099, 22.590012907313445170208991969752, 23.92704745497173991793947221541, 25.79510710457196133195397145340, 26.23859741709644107998707403878, 27.09398971682009675358241845409, 28.34794956762620432494760712220, 29.88338231914811949753897318146, 30.081847973308800794430706170719

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