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Properties

Label 1-5733-5733.3287-r0-0-0
Degree $1$
Conductor $5733$
Sign $-0.342 - 0.939i$
Analytic cond. $26.6239$
Root an. cond. $26.6239$
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  + (0.930 − 0.365i)2-s + (0.733 − 0.680i)4-s + (0.997 − 0.0747i)5-s + (0.433 − 0.900i)8-s + (0.900 − 0.433i)10-s + (0.149 − 0.988i)11-s + (0.0747 − 0.997i)16-s + (−0.733 − 0.680i)17-s + i·19-s + (0.680 − 0.733i)20-s + (−0.222 − 0.974i)22-s + (−0.222 − 0.974i)23-s + (0.988 − 0.149i)25-s + (0.222 − 0.974i)29-s + i·31-s + (−0.294 − 0.955i)32-s + ⋯
L(s)  = 1  + (0.930 − 0.365i)2-s + (0.733 − 0.680i)4-s + (0.997 − 0.0747i)5-s + (0.433 − 0.900i)8-s + (0.900 − 0.433i)10-s + (0.149 − 0.988i)11-s + (0.0747 − 0.997i)16-s + (−0.733 − 0.680i)17-s + i·19-s + (0.680 − 0.733i)20-s + (−0.222 − 0.974i)22-s + (−0.222 − 0.974i)23-s + (0.988 − 0.149i)25-s + (0.222 − 0.974i)29-s + i·31-s + (−0.294 − 0.955i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5733\)    =    \(3^{2} \cdot 7^{2} \cdot 13\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(26.6239\)
Root analytic conductor: \(26.6239\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5733} (3287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5733,\ (0:\ ),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.403725735 - 3.434948714i\)
\(L(\frac12)\) \(\approx\) \(2.403725735 - 3.434948714i\)
\(L(1)\) \(\approx\) \(1.970818255 - 0.9943452114i\)
\(L(1)\) \(\approx\) \(1.970818255 - 0.9943452114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.930 - 0.365i)T \)
5 \( 1 + (0.997 - 0.0747i)T \)
11 \( 1 + (0.149 - 0.988i)T \)
17 \( 1 + (-0.733 - 0.680i)T \)
19 \( 1 + iT \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + iT \)
37 \( 1 + (-0.294 - 0.955i)T \)
41 \( 1 + (0.997 - 0.0747i)T \)
43 \( 1 + (-0.0747 + 0.997i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (0.733 - 0.680i)T \)
59 \( 1 + (-0.563 + 0.826i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.680 + 0.733i)T \)
73 \( 1 + (-0.930 - 0.365i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.930 - 0.365i)T \)
89 \( 1 + (0.930 + 0.365i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−17.62494274700581640551870211128, −17.39340865976683103414120838213, −16.74696697107221351124604860395, −15.82670485886920523722903410374, −15.20200763746980593541931370085, −14.80710115236657863235941307554, −13.91350269440744018464088925322, −13.460664030516570454028677647, −12.9002871834139755163376420804, −12.25528325761999549770774501450, −11.48367257345734453262974380192, −10.75186752142494995763582126318, −10.10866481998590424775221156622, −9.22595187436146753600917057541, −8.66060978992600596473197099898, −7.60704646800636067556518803764, −6.990102216076969297984900581751, −6.44537021409502630678040278889, −5.6897726682750088965750237914, −5.0644483595807259872960153655, −4.381439181103700243671276748696, −3.63347528279189369414811367149, −2.6137990315499063946755715247, −2.12671010900748946321236179712, −1.32952783195679269171481450472, 0.68492325649068045526372844284, 1.52622381956707820446662748891, 2.409457123462738329143649680190, 2.86889652581466939158631341864, 3.858883519728817646826994989803, 4.52706775230316679988874617787, 5.35293430331271580613390982645, 5.9687779818224989715990854924, 6.41555316566700440457816238617, 7.213328203873516547208125006581, 8.22905436901190437163362297997, 9.05022750309642655118976405499, 9.69397921741487843269264494335, 10.51832738353000124151934343589, 10.88036069152463765807365213451, 11.75773807653249682691030464822, 12.3924292493692116519748792016, 13.08332306012014726318845648885, 13.659076720790429770218862929324, 14.26729991022223201876922268034, 14.5619359825790449888417244959, 15.65036482904701785308809931657, 16.259752577760873881435794860724, 16.690540988167692998479811881644, 17.70722425476342849230509508244

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