Properties

Label 1-1053-1053.283-r0-0-0
Degree 11
Conductor 10531053
Sign 0.761+0.647i0.761 + 0.647i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.396 + 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.893 − 0.448i)5-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (0.766 − 0.642i)10-s + (0.835 + 0.549i)11-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)16-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.286 + 0.957i)20-s + (−0.835 + 0.549i)22-s + (−0.686 − 0.727i)23-s + (0.597 + 0.802i)25-s + ⋯
L(s)  = 1  + (−0.396 + 0.918i)2-s + (−0.686 − 0.727i)4-s + (−0.893 − 0.448i)5-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (0.766 − 0.642i)10-s + (0.835 + 0.549i)11-s + (0.396 + 0.918i)14-s + (−0.0581 + 0.998i)16-s + (−0.939 − 0.342i)17-s + (−0.173 + 0.984i)19-s + (0.286 + 0.957i)20-s + (−0.835 + 0.549i)22-s + (−0.686 − 0.727i)23-s + (0.597 + 0.802i)25-s + ⋯

Functional equation

Λ(s)=(1053s/2ΓR(s)L(s)=((0.761+0.647i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1053s/2ΓR(s)L(s)=((0.761+0.647i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 10531053    =    34133^{4} \cdot 13
Sign: 0.761+0.647i0.761 + 0.647i
Analytic conductor: 4.890114.89011
Root analytic conductor: 4.890114.89011
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1053(283,)\chi_{1053} (283, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 1053, (0: ), 0.761+0.647i)(1,\ 1053,\ (0:\ ),\ 0.761 + 0.647i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.9378208814+0.3448313349i0.9378208814 + 0.3448313349i
L(12)L(\frac12) \approx 0.9378208814+0.3448313349i0.9378208814 + 0.3448313349i
L(1)L(1) \approx 0.7709803377+0.2177577893i0.7709803377 + 0.2177577893i
L(1)L(1) \approx 0.7709803377+0.2177577893i0.7709803377 + 0.2177577893i

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+(0.396+0.918i)T 1 + (-0.396 + 0.918i)T
5 1+(0.8930.448i)T 1 + (-0.893 - 0.448i)T
7 1+(0.6860.727i)T 1 + (0.686 - 0.727i)T
11 1+(0.835+0.549i)T 1 + (0.835 + 0.549i)T
17 1+(0.9390.342i)T 1 + (-0.939 - 0.342i)T
19 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
23 1+(0.6860.727i)T 1 + (-0.686 - 0.727i)T
29 1+(0.597+0.802i)T 1 + (0.597 + 0.802i)T
31 1+(0.686+0.727i)T 1 + (0.686 + 0.727i)T
37 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
41 1+(0.597+0.802i)T 1 + (-0.597 + 0.802i)T
43 1+(0.0581+0.998i)T 1 + (-0.0581 + 0.998i)T
47 1+(0.6860.727i)T 1 + (0.686 - 0.727i)T
53 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
59 1+(0.8350.549i)T 1 + (0.835 - 0.549i)T
61 1+(0.2860.957i)T 1 + (-0.286 - 0.957i)T
67 1+(0.993+0.116i)T 1 + (0.993 + 0.116i)T
71 1+(0.1730.984i)T 1 + (-0.173 - 0.984i)T
73 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
79 1+(0.993+0.116i)T 1 + (-0.993 + 0.116i)T
83 1+(0.9930.116i)T 1 + (0.993 - 0.116i)T
89 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
97 1+(0.835+0.549i)T 1 + (0.835 + 0.549i)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

−21.60098309169300492471013157649, −20.49060091802362873063809881601, −19.760069553316819518652843668841, −19.183254137558936987190342011995, −18.53912545600016931117828850955, −17.63746763059249477600160950685, −17.11795847205257962021047986684, −15.83007316682794241170469393401, −15.26435273248443013365796272643, −14.22767682236872834715856317313, −13.49485896984101213702931250579, −12.38571450949886465459569669304, −11.58956950217180791097230434432, −11.35508935100698386912890923190, −10.452117635416768516078642249990, −9.29490027554709157694800762252, −8.61433284353751380994932687267, −7.97101155861263630895837305118, −6.98112877665043908962994518050, −5.831427521461385082499934447055, −4.473774951354341674999181228525, −3.998439386478035116614087241494, −2.83155360809384327192212260346, −2.086320737193342177784291117458, −0.750322970293837274114390164831, 0.809109942373181210409112261779, 1.77363933285754488029086910974, 3.67610864316940762822732684171, 4.50617345828863583049401666244, 4.93598906496871939863371660002, 6.42881877869931034240334022186, 6.965853203335228422087247682972, 8.039796081367258683993662125164, 8.35208595197889090268122094953, 9.40419252638127449467826157669, 10.29806233740512237727992301481, 11.190747270670060470930227675866, 12.07979184503164683831100686179, 13.01984469955509579051801241020, 14.07860820341258462913287147626, 14.61018683067898427498441687345, 15.392968928195271599371614895090, 16.31055360932203086806910526241, 16.75949959925907619735836276566, 17.65404889684180572000424984542, 18.26335828297948087182853677387, 19.29851861773542524615583574213, 20.05171978560654079291725728469, 20.423806668748795830880699455674, 21.78329363454484193550316498555

Graph of the ZZ-function along the critical line