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Properties

Label 1-33e2-1089.128-r0-0-0
Degree $1$
Conductor $1089$
Sign $-0.956 - 0.291i$
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  + (0.123 + 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

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

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.956 - 0.291i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ -0.956 - 0.291i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08305392435 + 0.5582050669i\)
\(L(\frac12)\) \(\approx\) \(-0.08305392435 + 0.5582050669i\)
\(L(1)\) \(\approx\) \(0.6370973381 + 0.4393458835i\)
\(L(1)\) \(\approx\) \(0.6370973381 + 0.4393458835i\)

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

−20.87398016732352722063830051924, −20.27244185427229543068275365020, −19.41340607016039147963062624989, −19.04554594802311438473742922359, −18.0007629026909499089107766486, −17.37053615596655766369500709097, −16.42989230449273026397414661811, −15.29432277411613319075982285004, −14.65358665695992218908222241902, −13.80997568556356757668898297351, −13.14281218651810330297662151636, −11.92294321432636439114008607537, −11.64159663175499944694031940898, −10.70122325653056720127874844043, −10.25612142686509916970867890297, −8.77190949800044086331620027434, −8.48210988009138155860578933050, −7.29289961895159037808995092663, −6.44481892305039732387539129365, −4.89942035977319053219010994909, −4.357563721862652381946657947978, −3.68590291339170143095676476315, −2.50926333986592492963237612687, −1.562018838600291863544562408063, −0.2455879353861960656979038275, 1.29701470152861632780022616357, 2.92878001219804179303968792219, 3.90630749869511477243152211270, 4.81425540416739608486796415711, 5.45428417819324966506942537692, 6.41546121503056271947556777983, 7.63537850574075100266572241211, 7.93827672225255777292627831246, 8.77230029928970006365945854064, 9.58448391087658961007430380763, 10.8498135489104028084390329019, 11.72207574985228872687451173625, 12.50502376014973946954388858212, 13.2337514493724082613214145973, 14.32599901687456626130391840770, 14.9618817642800307130698624721, 15.63297907724755813922348527607, 16.16686991827311437378338280493, 17.144808652040440704392221694, 17.97595661344231376022775797340, 18.476277982353108569272821343415, 19.4429111747806353963736722559, 20.32312143150582758203833276275, 21.17329988336265627740837795403, 22.07760382705325512172946172608

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