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Properties

Label 1-33e2-1089.103-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.796 + 0.604i$
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  + (−0.935 + 0.353i)2-s + (0.749 − 0.662i)4-s + (−0.0665 − 0.997i)5-s + (0.820 + 0.572i)7-s + (−0.466 + 0.884i)8-s + (0.415 + 0.909i)10-s + (−0.398 − 0.917i)13-s + (−0.969 − 0.244i)14-s + (0.123 − 0.992i)16-s + (−0.564 + 0.825i)17-s + (0.941 + 0.336i)19-s + (−0.710 − 0.703i)20-s + (−0.327 + 0.945i)23-s + (−0.991 + 0.132i)25-s + (0.696 + 0.717i)26-s + ⋯
L(s)  = 1  + (−0.935 + 0.353i)2-s + (0.749 − 0.662i)4-s + (−0.0665 − 0.997i)5-s + (0.820 + 0.572i)7-s + (−0.466 + 0.884i)8-s + (0.415 + 0.909i)10-s + (−0.398 − 0.917i)13-s + (−0.969 − 0.244i)14-s + (0.123 − 0.992i)16-s + (−0.564 + 0.825i)17-s + (0.941 + 0.336i)19-s + (−0.710 − 0.703i)20-s + (−0.327 + 0.945i)23-s + (−0.991 + 0.132i)25-s + (0.696 + 0.717i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.796 + 0.604i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.796 + 0.604i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9356173549 + 0.3150803778i\)
\(L(\frac12)\) \(\approx\) \(0.9356173549 + 0.3150803778i\)
\(L(1)\) \(\approx\) \(0.7715576120 + 0.08267864093i\)
\(L(1)\) \(\approx\) \(0.7715576120 + 0.08267864093i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.935 + 0.353i)T \)
5 \( 1 + (-0.0665 - 0.997i)T \)
7 \( 1 + (0.820 + 0.572i)T \)
13 \( 1 + (-0.398 - 0.917i)T \)
17 \( 1 + (-0.564 + 0.825i)T \)
19 \( 1 + (0.941 + 0.336i)T \)
23 \( 1 + (-0.327 + 0.945i)T \)
29 \( 1 + (0.380 + 0.924i)T \)
31 \( 1 + (0.953 + 0.299i)T \)
37 \( 1 + (-0.870 - 0.491i)T \)
41 \( 1 + (0.449 + 0.893i)T \)
43 \( 1 + (0.928 + 0.371i)T \)
47 \( 1 + (0.988 + 0.151i)T \)
53 \( 1 + (-0.921 - 0.389i)T \)
59 \( 1 + (0.548 + 0.836i)T \)
61 \( 1 + (0.161 + 0.986i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (0.974 + 0.226i)T \)
79 \( 1 + (0.345 - 0.938i)T \)
83 \( 1 + (-0.999 + 0.0190i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.0665 + 0.997i)T \)
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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

−21.051222529541859081291085776658, −20.6019120024502519098335616794, −19.66669771252529172121482777963, −18.94410968526639906686186300187, −18.301539439482252296289304862209, −17.55820935795568119643128571418, −17.01744092798837424090981680005, −15.90443061049404838252962949625, −15.33898815500788993170264999167, −14.08009722885151707046782771230, −13.837076575198860044135835839161, −12.27285956770082762107794609580, −11.569229671206345034648410268243, −11.03284585600543426221837744726, −10.21168350434764593602369441830, −9.497762357883896052632152873939, −8.507865997257887345007971300889, −7.56243903389097761322244522868, −7.06107550310245225612827724548, −6.25019551521601610522006898746, −4.72590215616211600978658568564, −3.83065341802029968923744607029, −2.65760723069658070586679213315, −2.03963057398947306781361031369, −0.66975481968764837634468877765, 1.036741811837219364464819724565, 1.77593090012733887454249722248, 2.94679050238925659129345375834, 4.45643491688701436456579467873, 5.41542944629977470007170552370, 5.86339528540286201988404149054, 7.263930396066551991151728425261, 8.018803608758822775858293080275, 8.57529419815174066222483392814, 9.3504984784901897896823287589, 10.20955973622564289542267128312, 11.1245852019811501394038967789, 11.97377423621228233130515376664, 12.60407846418456011786212787752, 13.81673158706473899065858167215, 14.70506263312779393435511169011, 15.573278716229605531970206463647, 15.98144277356935630462977769280, 17.04705307171810393571374785380, 17.70677844548691106595448245401, 18.05570458531945900875276991524, 19.33872621274065016670993078931, 19.7879316930596677003286052057, 20.6409869635121149876265268171, 21.2100012481592315862068857984

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