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Properties

Label 1-33e2-1089.322-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.0135 - 0.999i$
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  + (−0.710 + 0.703i)2-s + (0.00951 − 0.999i)4-s + (0.345 − 0.938i)5-s + (−0.683 − 0.730i)7-s + (0.696 + 0.717i)8-s + (0.415 + 0.909i)10-s + (0.953 − 0.299i)13-s + (0.999 + 0.0380i)14-s + (−0.999 − 0.0190i)16-s + (−0.0285 − 0.999i)17-s + (0.610 − 0.791i)19-s + (−0.935 − 0.353i)20-s + (0.981 − 0.189i)23-s + (−0.761 − 0.647i)25-s + (−0.466 + 0.884i)26-s + ⋯
L(s)  = 1  + (−0.710 + 0.703i)2-s + (0.00951 − 0.999i)4-s + (0.345 − 0.938i)5-s + (−0.683 − 0.730i)7-s + (0.696 + 0.717i)8-s + (0.415 + 0.909i)10-s + (0.953 − 0.299i)13-s + (0.999 + 0.0380i)14-s + (−0.999 − 0.0190i)16-s + (−0.0285 − 0.999i)17-s + (0.610 − 0.791i)19-s + (−0.935 − 0.353i)20-s + (0.981 − 0.189i)23-s + (−0.761 − 0.647i)25-s + (−0.466 + 0.884i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0135 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0135 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.0135 - 0.999i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.0135 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6632776984 - 0.6723324432i\)
\(L(\frac12)\) \(\approx\) \(0.6632776984 - 0.6723324432i\)
\(L(1)\) \(\approx\) \(0.7670183574 - 0.1395417780i\)
\(L(1)\) \(\approx\) \(0.7670183574 - 0.1395417780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.710 + 0.703i)T \)
5 \( 1 + (0.345 - 0.938i)T \)
7 \( 1 + (-0.683 - 0.730i)T \)
13 \( 1 + (0.953 - 0.299i)T \)
17 \( 1 + (-0.0285 - 0.999i)T \)
19 \( 1 + (0.610 - 0.791i)T \)
23 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (-0.179 - 0.983i)T \)
31 \( 1 + (-0.398 + 0.917i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (-0.625 - 0.780i)T \)
43 \( 1 + (-0.786 + 0.618i)T \)
47 \( 1 + (0.548 + 0.836i)T \)
53 \( 1 + (0.516 + 0.856i)T \)
59 \( 1 + (0.988 + 0.151i)T \)
61 \( 1 + (0.964 - 0.263i)T \)
67 \( 1 + (0.0475 + 0.998i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (0.0855 + 0.996i)T \)
79 \( 1 + (-0.0665 - 0.997i)T \)
83 \( 1 + (0.123 + 0.992i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.345 + 0.938i)T \)
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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

−21.608156946032596270941320451557, −20.88030581622126762654609244453, −19.9734618871243254393066216210, −19.06136099009417585735692368127, −18.579603983931695020989282061777, −18.14403130988156935096128007089, −17.04855484691364622740675932655, −16.39628365923237814901505856027, −15.42595575636790618350387922916, −14.68463458686663540138813473398, −13.441403550455049538654512631089, −13.00238457614279741920518853093, −11.89494603244509039398296246692, −11.2827621414992347846374801324, −10.38848442992696841673566111032, −9.80338021548185651928052818958, −8.9187051284363681216197606139, −8.197432030947309182614056010807, −7.042352867134042516630800270122, −6.3876476291923281411107864652, −5.404366410547544029139605936717, −3.72344476498897775703377291505, −3.31414284955650154639381143455, −2.26296267246516453736465842301, −1.39629735162644642928225580353, 0.56445227051036215121590820513, 1.280123654884645999180344233069, 2.717621912047756097261212703518, 4.06434053749671396270385095259, 5.06521321501143291806152177724, 5.7753350320282309034254990524, 6.78737883610131191831207964324, 7.4296463968987145249046696221, 8.50679834128527108898967249389, 9.16673721899231247424182512685, 9.7964547160905140618060457038, 10.69861641326687540506526529314, 11.53933740299076333764234541738, 12.83860117691448507864430589061, 13.518696183752020539684454920189, 14.05416823152090453774975423744, 15.323471561313507439788684819815, 16.078564701333144992945679582127, 16.43410553859361466367779043571, 17.36929322819067215106473439853, 17.89160207892582324078770529575, 18.857825364949791401380681927662, 19.66955165332700135523128403223, 20.379661449654042916583091061427, 20.87033309409771512257182187320

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