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Properties

Label 1-33e2-1089.293-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.981 + 0.190i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (0.879 + 0.475i)2-s + (0.548 + 0.836i)4-s + (−0.483 + 0.875i)5-s + (0.948 + 0.318i)7-s + (0.0855 + 0.996i)8-s + (−0.841 + 0.540i)10-s + (−0.964 + 0.263i)13-s + (0.683 + 0.730i)14-s + (−0.398 + 0.917i)16-s + (−0.985 − 0.170i)17-s + (−0.696 + 0.717i)19-s + (−0.997 + 0.0760i)20-s + (−0.580 + 0.814i)23-s + (−0.532 − 0.846i)25-s + (−0.974 − 0.226i)26-s + ⋯
L(s)  = 1  + (0.879 + 0.475i)2-s + (0.548 + 0.836i)4-s + (−0.483 + 0.875i)5-s + (0.948 + 0.318i)7-s + (0.0855 + 0.996i)8-s + (−0.841 + 0.540i)10-s + (−0.964 + 0.263i)13-s + (0.683 + 0.730i)14-s + (−0.398 + 0.917i)16-s + (−0.985 − 0.170i)17-s + (−0.696 + 0.717i)19-s + (−0.997 + 0.0760i)20-s + (−0.580 + 0.814i)23-s + (−0.532 − 0.846i)25-s + (−0.974 − 0.226i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.981 + 0.190i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.981 + 0.190i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1835636925 + 1.910714962i\)
\(L(\frac12)\) \(\approx\) \(0.1835636925 + 1.910714962i\)
\(L(1)\) \(\approx\) \(1.160247732 + 0.9707961756i\)
\(L(1)\) \(\approx\) \(1.160247732 + 0.9707961756i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.879 + 0.475i)T \)
5 \( 1 + (-0.483 + 0.875i)T \)
7 \( 1 + (0.948 + 0.318i)T \)
13 \( 1 + (-0.964 + 0.263i)T \)
17 \( 1 + (-0.985 - 0.170i)T \)
19 \( 1 + (-0.696 + 0.717i)T \)
23 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (0.999 + 0.0380i)T \)
31 \( 1 + (0.161 - 0.986i)T \)
37 \( 1 + (-0.362 + 0.931i)T \)
41 \( 1 + (0.380 - 0.924i)T \)
43 \( 1 + (-0.981 + 0.189i)T \)
47 \( 1 + (0.761 - 0.647i)T \)
53 \( 1 + (-0.993 - 0.113i)T \)
59 \( 1 + (0.991 + 0.132i)T \)
61 \( 1 + (0.851 + 0.524i)T \)
67 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (-0.897 - 0.441i)T \)
73 \( 1 + (0.870 + 0.491i)T \)
79 \( 1 + (-0.797 + 0.603i)T \)
83 \( 1 + (0.953 - 0.299i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.483 + 0.875i)T \)
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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

−21.16290302629642531063686698696, −20.18902979707909302377693740963, −19.89669404437127684233440602906, −19.12999254775411861383871894488, −17.857609412308152171912162823193, −17.23075684032605421785558473924, −16.17063095195245706619844643675, −15.48767998446340456419148131859, −14.680699767263800613117284111870, −14.006477603333536574325645838121, −13.03206972345084141828494767337, −12.44133453063413321902417647217, −11.69278273883054818745627150035, −10.92836837120416723029836576120, −10.17997729270345053727307106707, −9.014527521847013122009917545308, −8.19935869389775213560769176565, −7.21511195868959679782230067667, −6.28904040496805189620605283487, −4.93686144339676344327775789254, −4.75616080595031045342276844547, −3.902162113801982906227418679434, −2.59260230205385900868984068329, −1.7336530193314802890418411455, −0.53854900485832160752846254695, 1.9996736640112626530217169859, 2.60155003193139329725604974791, 3.82674503661290127394710954843, 4.492754829374624129605662944071, 5.408993895486905405583290734847, 6.405307447438741235771039043151, 7.13851978272166258922746788370, 7.93387405891916660727406900237, 8.59736599108145638169928058232, 10.02147798037463839838587620917, 10.97414039941928804372818193954, 11.74271984093077639070249559739, 12.14194680129571527297939772613, 13.363750167068446089512673335891, 14.1521817974866471401896062667, 14.77907256755789532055620363978, 15.32167548762522975017032321082, 16.059977198085230155889864251168, 17.22142101107808252794283011289, 17.67910954226138479994721901040, 18.6870651329056537746904461945, 19.5534234075269503923120894268, 20.41412732238343308956240383033, 21.25800454200375657851150064759, 22.039575201928202968333736673656

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