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Properties

Label 1-33e2-1089.311-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.291 - 0.956i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (−0.123 + 0.992i)2-s + (−0.969 − 0.244i)4-s + (−0.964 + 0.263i)5-s + (−0.761 + 0.647i)7-s + (0.362 − 0.931i)8-s + (−0.142 − 0.989i)10-s + (−0.0665 − 0.997i)13-s + (−0.548 − 0.836i)14-s + (0.879 + 0.475i)16-s + (0.736 − 0.676i)17-s + (0.198 + 0.980i)19-s + (0.999 − 0.0190i)20-s + (−0.235 − 0.971i)23-s + (0.861 − 0.508i)25-s + (0.998 + 0.0570i)26-s + ⋯
L(s)  = 1  + (−0.123 + 0.992i)2-s + (−0.969 − 0.244i)4-s + (−0.964 + 0.263i)5-s + (−0.761 + 0.647i)7-s + (0.362 − 0.931i)8-s + (−0.142 − 0.989i)10-s + (−0.0665 − 0.997i)13-s + (−0.548 − 0.836i)14-s + (0.879 + 0.475i)16-s + (0.736 − 0.676i)17-s + (0.198 + 0.980i)19-s + (0.999 − 0.0190i)20-s + (−0.235 − 0.971i)23-s + (0.861 − 0.508i)25-s + (0.998 + 0.0570i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.291 - 0.956i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.291 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1329265459 + 0.1793962702i\)
\(L(\frac12)\) \(\approx\) \(-0.1329265459 + 0.1793962702i\)
\(L(1)\) \(\approx\) \(0.5397304670 + 0.3555831660i\)
\(L(1)\) \(\approx\) \(0.5397304670 + 0.3555831660i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.123 + 0.992i)T \)
5 \( 1 + (-0.964 + 0.263i)T \)
7 \( 1 + (-0.761 + 0.647i)T \)
13 \( 1 + (-0.0665 - 0.997i)T \)
17 \( 1 + (0.736 - 0.676i)T \)
19 \( 1 + (0.198 + 0.980i)T \)
23 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 + (-0.00951 + 0.999i)T \)
31 \( 1 + (0.345 + 0.938i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (0.290 + 0.956i)T \)
43 \( 1 + (0.0475 + 0.998i)T \)
47 \( 1 + (-0.820 - 0.572i)T \)
53 \( 1 + (0.0285 - 0.999i)T \)
59 \( 1 + (0.683 + 0.730i)T \)
61 \( 1 + (0.797 - 0.603i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.993 - 0.113i)T \)
73 \( 1 + (0.610 + 0.791i)T \)
79 \( 1 + (0.161 + 0.986i)T \)
83 \( 1 + (-0.997 + 0.0760i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.964 + 0.263i)T \)
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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

−20.63744175287166549617170293304, −19.7506528403896878571766172643, −19.253241118455870665017716774141, −18.87570093107167561756967453423, −17.55180646973274501451506330325, −16.946444440260563326683181625741, −16.12524577181843859107602571415, −15.265719480412176825572822361021, −14.14269708222650586695250086280, −13.44837012888082748673415980940, −12.66646793663768726572339941518, −11.88305460712322378305957789407, −11.28603376952270228066620681895, −10.39540992535553505018591600068, −9.531552882605722632879701256933, −8.880340724568202642997970675178, −7.802252033211572423370302475254, −7.17298983199609276559650919799, −5.86198393956496354342403159832, −4.63572770153585364355534344703, −3.92261343989039956697467381937, −3.34061965662633575319498170955, −2.11367462454801831790490688589, −0.893853806639546665318801594296, −0.07272602146292936998038288637, 1.02486017544695237107321789529, 2.95405681344647033137995641069, 3.53161415185340166319384991675, 4.75493061328714278742466333799, 5.53925880569804004507861226870, 6.48407748504302852692523799521, 7.197396862109917928987690987348, 8.16738428112492878530143437214, 8.58462355283685583482933721618, 9.848081344726373623119976420814, 10.31057673505845160361817111853, 11.66926805349284316656694828149, 12.4956234507829999303479959470, 13.05570054520252768251632806429, 14.42250151640259474601733149843, 14.7153490994727662729338020704, 15.80482525263789449062393433844, 16.093291789209244095478429564589, 16.84631067507254008436961809676, 18.221779237510480880564564580370, 18.368446699607207079734616076797, 19.33401457162331817086495087088, 19.95899982502500450876936421355, 21.07961111556378106415807886834, 22.26861809289211738521142236341

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