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Properties

Label 1-33e2-1089.301-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: $\chi_{1089} (301, \cdot )$
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

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

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