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Properties

Label 1-1053-1053.160-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.761 - 0.647i$
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 + (−0.893 + 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

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

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.761 - 0.647i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.761 - 0.647i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9378208814 - 0.3448313349i\)
\(L(\frac12)\) \(\approx\) \(0.9378208814 - 0.3448313349i\)
\(L(1)\) \(\approx\) \(0.7709803377 - 0.2177577893i\)
\(L(1)\) \(\approx\) \(0.7709803377 - 0.2177577893i\)

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

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

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