Properties

Label 1-1053-1053.256-r0-0-0
Degree 11
Conductor 10531053
Sign 0.9970.0742i0.997 - 0.0742i
Analytic cond. 4.890114.89011
Root an. cond. 4.890114.89011
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(1053s/2ΓR(s)L(s)=((0.9970.0742i)Λ(1s)\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}
Λ(s)=(1053s/2ΓR(s)L(s)=((0.9970.0742i)Λ(1s)\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: 11
Conductor: 10531053    =    34133^{4} \cdot 13
Sign: 0.9970.0742i0.997 - 0.0742i
Analytic conductor: 4.890114.89011
Root analytic conductor: 4.890114.89011
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1053(256,)\chi_{1053} (256, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1053, (0: ), 0.9970.0742i)(1,\ 1053,\ (0:\ ),\ 0.997 - 0.0742i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.7927017229+0.02945676966i0.7927017229 + 0.02945676966i
L(12)L(\frac12) \approx 0.7927017229+0.02945676966i0.7927017229 + 0.02945676966i
L(1)L(1) \approx 0.76395984940.3464779210i0.7639598494 - 0.3464779210i
L(1)L(1) \approx 0.76395984940.3464779210i0.7639598494 - 0.3464779210i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
13 1 1
good2 1+(0.3960.918i)T 1 + (0.396 - 0.918i)T
5 1+(0.835+0.549i)T 1 + (-0.835 + 0.549i)T
7 1+(0.2860.957i)T 1 + (-0.286 - 0.957i)T
11 1+(0.8350.549i)T 1 + (-0.835 - 0.549i)T
17 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
19 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
23 1+(0.286+0.957i)T 1 + (-0.286 + 0.957i)T
29 1+(0.597+0.802i)T 1 + (0.597 + 0.802i)T
31 1+(0.9730.230i)T 1 + (0.973 - 0.230i)T
37 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
41 1+(0.9930.116i)T 1 + (-0.993 - 0.116i)T
43 1+(0.8350.549i)T 1 + (-0.835 - 0.549i)T
47 1+(0.973+0.230i)T 1 + (0.973 + 0.230i)T
53 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
59 1+(0.835+0.549i)T 1 + (-0.835 + 0.549i)T
61 1+(0.973+0.230i)T 1 + (0.973 + 0.230i)T
67 1+(0.396+0.918i)T 1 + (0.396 + 0.918i)T
71 1+(0.7660.642i)T 1 + (0.766 - 0.642i)T
73 1+(0.939+0.342i)T 1 + (-0.939 + 0.342i)T
79 1+(0.597+0.802i)T 1 + (0.597 + 0.802i)T
83 1+(0.597+0.802i)T 1 + (0.597 + 0.802i)T
89 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
97 1+(0.8930.448i)T 1 + (0.893 - 0.448i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line