↑

Properties

Label 1-1053-1053.259-r0-0-0
Degree $1$
Conductor $1053$
Sign $-0.360 - 0.932i$
Analytic cond. $4.89011$
Root an. cond. $4.89011$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.893 − 0.448i)2-s + (0.597 + 0.802i)4-s + (0.286 + 0.957i)5-s + (−0.396 − 0.918i)7-s + (−0.173 − 0.984i)8-s + (0.173 − 0.984i)10-s + (−0.973 + 0.230i)11-s + (−0.0581 + 0.998i)14-s + (−0.286 + 0.957i)16-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)19-s + (−0.597 + 0.802i)20-s + (0.973 + 0.230i)22-s + (0.396 − 0.918i)23-s + (−0.835 + 0.549i)25-s + ⋯
L(s)  = 1  + (−0.893 − 0.448i)2-s + (0.597 + 0.802i)4-s + (0.286 + 0.957i)5-s + (−0.396 − 0.918i)7-s + (−0.173 − 0.984i)8-s + (0.173 − 0.984i)10-s + (−0.973 + 0.230i)11-s + (−0.0581 + 0.998i)14-s + (−0.286 + 0.957i)16-s + (−0.939 + 0.342i)17-s + (0.939 + 0.342i)19-s + (−0.597 + 0.802i)20-s + (0.973 + 0.230i)22-s + (0.396 − 0.918i)23-s + (−0.835 + 0.549i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.360 - 0.932i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ -0.360 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2889620096 - 0.4213169195i\)
\(L(\frac12)\) \(\approx\) \(0.2889620096 - 0.4213169195i\)
\(L(1)\) \(\approx\) \(0.5960978694 - 0.1159639094i\)
\(L(1)\) \(\approx\) \(0.5960978694 - 0.1159639094i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.893 - 0.448i)T \)
5 \( 1 + (0.286 + 0.957i)T \)
7 \( 1 + (-0.396 - 0.918i)T \)
11 \( 1 + (-0.973 + 0.230i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (0.396 - 0.918i)T \)
29 \( 1 + (-0.0581 - 0.998i)T \)
31 \( 1 + (0.993 - 0.116i)T \)
37 \( 1 + (-0.766 - 0.642i)T \)
41 \( 1 + (-0.893 + 0.448i)T \)
43 \( 1 + (-0.686 - 0.727i)T \)
47 \( 1 + (0.993 + 0.116i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.973 - 0.230i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.0581 - 0.998i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (0.893 + 0.448i)T \)
83 \( 1 + (-0.893 - 0.448i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.286 - 0.957i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.60929284614647146359090657250, −20.808513590684620601445188543238, −20.111289362977504252820383179920, −19.348891709212281459794935851487, −18.497953966408155557585655583425, −17.870603528703209978934187382552, −17.16926842231379615110501134030, −16.12191252518466896053955540325, −15.79240591037043708008914340942, −15.12453553025831563053616832692, −13.78641958400443375692243901135, −13.193512041632895489641547189244, −12.100017559796056616536062083895, −11.414677419023152633818858261343, −10.30832873303580780386557789544, −9.525006620695878522182418196614, −8.85651412901165436102552902227, −8.28191912458688485110277492488, −7.244458393869723627541977983884, −6.304695071002329882026654141774, −5.29254823408147896066472226677, −5.00329387913820030094252498886, −3.11831753989899192873786340482, −2.19682545781552155799337168904, −1.11294611187801328994497920632, 0.3081289250673970491522099183, 1.78563005461039976988597604464, 2.71226921911218021875046436436, 3.45420783426251300719186227986, 4.53190435964031569174378841924, 6.05135127436864158767831926980, 6.863606852157871267866188634205, 7.49256178776774779651050258838, 8.32510182070806531170591286932, 9.48860919554391484602223008316, 10.22986583584314575621494587054, 10.6341917466026341538902143333, 11.4383858485401786829379257722, 12.48687709975199239377781052436, 13.3806789884334019712283372373, 13.99512360582397753012395189029, 15.342971124192248599111961683763, 15.76963369951504136841921722727, 16.93665848312749464696852178995, 17.42024254570870211363617852749, 18.33979971244451755543105220672, 18.79499915015394423416056251328, 19.67843953810055813709312375314, 20.44828463934742990562838640800, 21.03270644757522857898480327398

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