L-function 1-1053-1053.259-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	3	$1$
	13	$1$
good	2	$1 + (-0.893 - 0.448i)T$
	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	$1 + (0.993 - 0.116i)T$

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