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Properties

Label 1-6027-6027.3503-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.999 + 0.0408i$
Analytic cond. $27.9892$
Root an. cond. $27.9892$
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  + (0.337 − 0.941i)2-s + (−0.772 − 0.635i)4-s + (0.575 − 0.817i)5-s + (−0.858 + 0.512i)8-s + (−0.575 − 0.817i)10-s + (0.925 − 0.379i)11-s + (−0.983 + 0.178i)13-s + (0.193 + 0.981i)16-s + (−0.163 − 0.986i)17-s + (−0.913 + 0.406i)19-s + (−0.963 + 0.266i)20-s + (−0.0448 − 0.998i)22-s + (−0.712 + 0.701i)23-s + (−0.337 − 0.941i)25-s + (−0.163 + 0.986i)26-s + ⋯
L(s)  = 1  + (0.337 − 0.941i)2-s + (−0.772 − 0.635i)4-s + (0.575 − 0.817i)5-s + (−0.858 + 0.512i)8-s + (−0.575 − 0.817i)10-s + (0.925 − 0.379i)11-s + (−0.983 + 0.178i)13-s + (0.193 + 0.981i)16-s + (−0.163 − 0.986i)17-s + (−0.913 + 0.406i)19-s + (−0.963 + 0.266i)20-s + (−0.0448 − 0.998i)22-s + (−0.712 + 0.701i)23-s + (−0.337 − 0.941i)25-s + (−0.163 + 0.986i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.999 + 0.0408i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (3503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.999 + 0.0408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9973720180 + 0.02039042606i\)
\(L(\frac12)\) \(\approx\) \(0.9973720180 + 0.02039042606i\)
\(L(1)\) \(\approx\) \(0.8808242439 - 0.5867417448i\)
\(L(1)\) \(\approx\) \(0.8808242439 - 0.5867417448i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.337 - 0.941i)T \)
5 \( 1 + (0.575 - 0.817i)T \)
11 \( 1 + (0.925 - 0.379i)T \)
13 \( 1 + (-0.983 + 0.178i)T \)
17 \( 1 + (-0.163 - 0.986i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.712 + 0.701i)T \)
29 \( 1 + (0.550 + 0.834i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.998 - 0.0598i)T \)
43 \( 1 + (-0.393 - 0.919i)T \)
47 \( 1 + (-0.337 + 0.941i)T \)
53 \( 1 + (0.772 + 0.635i)T \)
59 \( 1 + (-0.599 + 0.800i)T \)
61 \( 1 + (-0.251 + 0.967i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (0.550 - 0.834i)T \)
73 \( 1 + (-0.826 - 0.563i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.646 + 0.762i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−17.5814954254894512009852749854, −16.95830326760141967381484108771, −16.65050726550266897561766020400, −15.44006916474099245711424334935, −15.03128180691752957818369484779, −14.49832056439469614874448722672, −14.09335901296302712496066115594, −13.06631829549082138188026790596, −12.78620746017144122424845450284, −11.851509751149534526932309624710, −11.19086192693595029197731981046, −10.02736055829202753081292688306, −9.88191604492999595214996609741, −8.91801395137559648104644112917, −8.23832056445713348310160945700, −7.45811244490165652606438989300, −6.80628506881685004024310621747, −6.218229027434565328642356186, −5.8142281857849261460156645889, −4.65746212376783136400060692911, −4.24155066395182873590496116333, −3.36650966525645721407518164065, −2.470602367067127336722083281572, −1.79744795787454515161365742532, −0.23123515831866946142361025714, 0.96926334707977331348699559554, 1.62403126459551307471802680687, 2.36943140323246249364674283038, 3.15071820178950789612640880009, 4.13790506256083565591605334399, 4.58103007485001337997705099433, 5.37715049289580448415666447442, 5.95829008308796718812896689401, 6.78000450703856921895257764777, 7.773160519321812271408576555197, 8.8654604051935838238165683956, 9.01834219185606762382487905904, 9.81138288455774978628535229227, 10.36822169912362090620396119768, 11.21023691453740229611532998598, 12.078149575034878259326705033929, 12.20123515944352412694375676546, 13.07451555439022867974616914466, 13.72207683001505210548086499627, 14.26261209742969018397575000428, 14.77383568284119422627301950224, 15.75692034612530217507574108884, 16.644043116517322460696393695229, 17.006394059354975182635338788933, 17.92082855099165193223630910781

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