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Properties

Label 1-1323-1323.1031-r1-0-0
Degree $1$
Conductor $1323$
Sign $-0.522 + 0.852i$
Analytic cond. $142.176$
Root an. cond. $142.176$
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  + (−0.698 + 0.715i)2-s + (−0.0249 − 0.999i)4-s + (−0.542 + 0.840i)5-s + (0.733 + 0.680i)8-s + (−0.222 − 0.974i)10-s + (−0.698 + 0.715i)11-s + (−0.583 − 0.811i)13-s + (−0.998 + 0.0498i)16-s + (−0.623 − 0.781i)17-s + 19-s + (0.853 + 0.521i)20-s + (−0.0249 − 0.999i)22-s + (−0.878 − 0.478i)23-s + (−0.411 − 0.911i)25-s + (0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (−0.698 + 0.715i)2-s + (−0.0249 − 0.999i)4-s + (−0.542 + 0.840i)5-s + (0.733 + 0.680i)8-s + (−0.222 − 0.974i)10-s + (−0.698 + 0.715i)11-s + (−0.583 − 0.811i)13-s + (−0.998 + 0.0498i)16-s + (−0.623 − 0.781i)17-s + 19-s + (0.853 + 0.521i)20-s + (−0.0249 − 0.999i)22-s + (−0.878 − 0.478i)23-s + (−0.411 − 0.911i)25-s + (0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.522 + 0.852i$
Analytic conductor: \(142.176\)
Root analytic conductor: \(142.176\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (1:\ ),\ -0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2697283191 + 0.4816010378i\)
\(L(\frac12)\) \(\approx\) \(0.2697283191 + 0.4816010378i\)
\(L(1)\) \(\approx\) \(0.5314900681 + 0.2054995600i\)
\(L(1)\) \(\approx\) \(0.5314900681 + 0.2054995600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.698 + 0.715i)T \)
5 \( 1 + (-0.542 + 0.840i)T \)
11 \( 1 + (-0.698 + 0.715i)T \)
13 \( 1 + (-0.583 - 0.811i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.878 - 0.478i)T \)
29 \( 1 + (-0.878 + 0.478i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.988 + 0.149i)T \)
41 \( 1 + (0.998 + 0.0498i)T \)
43 \( 1 + (0.456 - 0.889i)T \)
47 \( 1 + (0.969 + 0.246i)T \)
53 \( 1 + (-0.365 - 0.930i)T \)
59 \( 1 + (-0.921 + 0.388i)T \)
61 \( 1 + (-0.0249 + 0.999i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.988 + 0.149i)T \)
73 \( 1 + (0.0747 + 0.997i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.583 - 0.811i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−20.39111526276651156691002486782, −19.63749489013378685286109173842, −19.17371658821887557231801344363, −18.34576945765833018054921088657, −17.4301230361620082804308344286, −16.85028274478474138894106255442, −15.948834153687783506388887808133, −15.61625793751831363447849818912, −14.09075041535235925737246900641, −13.389306226445079955256404943946, −12.540657289791177936729919631807, −11.91263168264624638417732124190, −11.21655982973782672692195139188, −10.403221158752080961133166036952, −9.3830555558974048237776936946, −8.89885661876260780670003666583, −7.90189620886744893121001534006, −7.520252422032804185675468790137, −6.18926000024325573749221898920, −5.02488203660759469593645906458, −4.169526768356538357701146855122, −3.37269176930182763270031895506, −2.25118657230505613099353001439, −1.31374581137320020593297661625, −0.2437014688380240760385813433, 0.56534285812082621346456750979, 2.1195040220905553882148209864, 2.87913596188894015141855389198, 4.24047160395075709936538189404, 5.15905404334812360050434388233, 5.99152080836052906862245371930, 7.13366378953672134218081535943, 7.43639401791002586942607604416, 8.19453006713970078066126726091, 9.30173536806357476226974427640, 10.08883118332276095689156606236, 10.64493545002785390527377370633, 11.540972049528580062962589215738, 12.43001158208512572092204544642, 13.61930031165900974277653544874, 14.30525207009464881653668895021, 15.180158986974625946177059681791, 15.60658480627322767223055549952, 16.27366295437840849648559812603, 17.42942257999652973153710628733, 17.97705306805991941124037450617, 18.52366225281074419825790793092, 19.31348138935184115283534345137, 20.2015342502030355651141744723, 20.56356720674734543396338075289

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