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Properties

Label 1-4563-4563.49-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.805 - 0.592i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
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  + (0.589 + 0.807i)2-s + (−0.303 + 0.952i)4-s + (0.0938 + 0.995i)5-s + (0.147 + 0.989i)7-s + (−0.948 + 0.316i)8-s + (−0.748 + 0.663i)10-s + (−0.830 + 0.556i)11-s + (−0.711 + 0.702i)14-s + (−0.815 − 0.579i)16-s + (0.948 − 0.316i)17-s − 19-s + (−0.977 − 0.213i)20-s + (−0.939 − 0.342i)22-s + (0.766 − 0.642i)23-s + (−0.982 + 0.186i)25-s + ⋯
L(s)  = 1  + (0.589 + 0.807i)2-s + (−0.303 + 0.952i)4-s + (0.0938 + 0.995i)5-s + (0.147 + 0.989i)7-s + (−0.948 + 0.316i)8-s + (−0.748 + 0.663i)10-s + (−0.830 + 0.556i)11-s + (−0.711 + 0.702i)14-s + (−0.815 − 0.579i)16-s + (0.948 − 0.316i)17-s − 19-s + (−0.977 − 0.213i)20-s + (−0.939 − 0.342i)22-s + (0.766 − 0.642i)23-s + (−0.982 + 0.186i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002594473776 + 0.0008508032416i\)
\(L(\frac12)\) \(\approx\) \(0.002594473776 + 0.0008508032416i\)
\(L(1)\) \(\approx\) \(0.7420923972 + 0.7685628382i\)
\(L(1)\) \(\approx\) \(0.7420923972 + 0.7685628382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.589 + 0.807i)T \)
5 \( 1 + (0.0938 + 0.995i)T \)
7 \( 1 + (0.147 + 0.989i)T \)
11 \( 1 + (-0.830 + 0.556i)T \)
17 \( 1 + (0.948 - 0.316i)T \)
19 \( 1 - T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.589 - 0.807i)T \)
31 \( 1 + (-0.653 + 0.757i)T \)
37 \( 1 + (0.354 - 0.935i)T \)
41 \( 1 + (0.452 - 0.891i)T \)
43 \( 1 + (-0.982 + 0.186i)T \)
47 \( 1 + (0.673 - 0.739i)T \)
53 \( 1 + (-0.748 - 0.663i)T \)
59 \( 1 + (0.815 - 0.579i)T \)
61 \( 1 + (-0.999 + 0.0268i)T \)
67 \( 1 + (0.991 - 0.133i)T \)
71 \( 1 + (-0.799 - 0.600i)T \)
73 \( 1 + (-0.278 + 0.960i)T \)
79 \( 1 + (-0.303 - 0.952i)T \)
83 \( 1 + (-0.998 - 0.0536i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.909 + 0.416i)T \)
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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

−17.604420986938834874303915494486, −16.8403640073306956873666616459, −16.45682744292152207502008585664, −15.46791257403535207635874331115, −14.80796443348278839886965152294, −14.055708428465560167680757088184, −13.31550171341955098863953026477, −12.992401084309203062464058552205, −12.41124283635678437594880347699, −11.40473411634787811390160627678, −10.9619604517985046405425898763, −10.19731376531896668908193497477, −9.623202323680824792725605352331, −8.78487237692031940248609706674, −8.06559138360236965675911738810, −7.29818394386954093734772425915, −6.17156926426004602585220944999, −5.544381471843432778650304210132, −4.86931814930467182356829474865, −4.21433687533266743114824005287, −3.49610243484120178352967024629, −2.7119057094728604984905176038, −1.547777267295458854360031860248, −1.12203985685357229475984096010, −0.000583931557386782496966370372, 2.01200312324906547493063118194, 2.562496972086094657291941569888, 3.279295620760767259973087731171, 4.126801728733055297975427453468, 5.11202212104567530008299375068, 5.56721564666844629573772788069, 6.32521096796324299935650849355, 7.03697215877942873361479241715, 7.65358910867034061772063212405, 8.346488053146147144575775024352, 9.12488683890588496271031771187, 9.91304159987903729234667347362, 10.77048060049824126960777921461, 11.46170330799387842672774477709, 12.32489911272916763275336529743, 12.757422401305145296899256423332, 13.52257557112863960812418928127, 14.480757875846175029718309129262, 14.7367191037367027022530976794, 15.34943296540565420171833432286, 15.93569438635467074419192222524, 16.724200914207148534712645675046, 17.54089312458857022135613466059, 18.06462976548772013190239577779, 18.73831738727772835712629920381

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