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Properties

Label 1-1053-1053.256-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.997 - 0.0742i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.396 − 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.835 + 0.549i)5-s + (−0.286 − 0.957i)7-s + (−0.939 + 0.342i)8-s + (0.173 + 0.984i)10-s + (−0.835 − 0.549i)11-s + (−0.993 − 0.116i)14-s + (−0.0581 + 0.998i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (0.973 + 0.230i)20-s + (−0.835 + 0.549i)22-s + (−0.286 + 0.957i)23-s + (0.396 − 0.918i)25-s + ⋯
L(s)  = 1  + (0.396 − 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.835 + 0.549i)5-s + (−0.286 − 0.957i)7-s + (−0.939 + 0.342i)8-s + (0.173 + 0.984i)10-s + (−0.835 − 0.549i)11-s + (−0.993 − 0.116i)14-s + (−0.0581 + 0.998i)16-s + (0.173 + 0.984i)17-s + (−0.939 + 0.342i)19-s + (0.973 + 0.230i)20-s + (−0.835 + 0.549i)22-s + (−0.286 + 0.957i)23-s + (0.396 − 0.918i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.997 - 0.0742i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.997 - 0.0742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7927017229 + 0.02945676966i\)
\(L(\frac12)\) \(\approx\) \(0.7927017229 + 0.02945676966i\)
\(L(1)\) \(\approx\) \(0.7639598494 - 0.3464779210i\)
\(L(1)\) \(\approx\) \(0.7639598494 - 0.3464779210i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.396 - 0.918i)T \)
5 \( 1 + (-0.835 + 0.549i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (-0.835 - 0.549i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.286 + 0.957i)T \)
29 \( 1 + (0.597 + 0.802i)T \)
31 \( 1 + (0.973 - 0.230i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.993 - 0.116i)T \)
43 \( 1 + (-0.835 - 0.549i)T \)
47 \( 1 + (0.973 + 0.230i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.835 + 0.549i)T \)
61 \( 1 + (0.973 + 0.230i)T \)
67 \( 1 + (0.396 + 0.918i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.597 + 0.802i)T \)
83 \( 1 + (0.597 + 0.802i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.893 - 0.448i)T \)
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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

−21.659133315203937823749015401299, −20.86898153424795084141095775925, −20.11120269725708097059876340595, −18.87150126865869035885905043409, −18.5014931635935255884147443877, −17.46931553246843005768370783071, −16.63831925057763647080160321338, −15.81552476513245863011370398683, −15.44507332912996714466868868085, −14.75185170633614387403639864220, −13.61159933592044282703588797842, −12.838929736740380020011009499776, −12.20887051616303239773500454794, −11.58177441448315576075595315684, −10.1427055863978154892039374300, −9.19387256477221023739823400777, −8.3768858978927149698108449854, −7.873287489640447910380392128235, −6.81198930305845398644133293113, −6.030025458527297423007349102945, −4.783775960788805420489156949527, −4.67523604422878174711904071908, −3.25082099412952029722663558214, −2.41559501316370408772088727448, −0.37428225177118964299251300998, 0.908865328895285965968698871903, 2.240004215347327176168935281606, 3.33862936977010739072566946589, 3.815258293976315380357833502304, 4.7205914406972374571514065423, 5.88132332654578494701483280531, 6.76732096890399446919692102016, 7.90567139543797564957669850821, 8.54511547375030214063348231485, 9.93453711890508124059191628598, 10.52220215533283597896590096458, 11.03257816627349767623693475102, 11.96026530760224191163514641369, 12.798043667764141853577465005068, 13.51246291523297173655362638474, 14.289385533669502160159578735941, 15.108941042283771784037970993203, 15.85765677129070581684113584357, 16.89273232106539486706039693100, 17.82505605816865837057377117845, 18.75649994284517161771080737560, 19.31332271833814812854892801419, 19.87130245444905826936636789546, 20.71455605650712114512434037057, 21.5310527123335681309390278354

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