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Properties

Label 1-6027-6027.347-r0-0-0
Degree $1$
Conductor $6027$
Sign $0.0425 - 0.999i$
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  + (−0.967 − 0.251i)2-s + (0.873 + 0.486i)4-s + (−0.611 + 0.791i)5-s + (−0.722 − 0.691i)8-s + (0.791 − 0.611i)10-s + (−0.316 − 0.948i)11-s + (0.493 − 0.869i)13-s + (0.525 + 0.850i)16-s + (0.717 − 0.696i)17-s + (−0.629 + 0.777i)19-s + (−0.919 + 0.393i)20-s + (0.0672 + 0.997i)22-s + (−0.599 − 0.800i)23-s + (−0.251 − 0.967i)25-s + (−0.696 + 0.717i)26-s + ⋯
L(s)  = 1  + (−0.967 − 0.251i)2-s + (0.873 + 0.486i)4-s + (−0.611 + 0.791i)5-s + (−0.722 − 0.691i)8-s + (0.791 − 0.611i)10-s + (−0.316 − 0.948i)11-s + (0.493 − 0.869i)13-s + (0.525 + 0.850i)16-s + (0.717 − 0.696i)17-s + (−0.629 + 0.777i)19-s + (−0.919 + 0.393i)20-s + (0.0672 + 0.997i)22-s + (−0.599 − 0.800i)23-s + (−0.251 − 0.967i)25-s + (−0.696 + 0.717i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0425 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0425 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $0.0425 - 0.999i$
Analytic conductor: \(27.9892\)
Root analytic conductor: \(27.9892\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6027} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6027,\ (0:\ ),\ 0.0425 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5279001203 - 0.5059010944i\)
\(L(\frac12)\) \(\approx\) \(0.5279001203 - 0.5059010944i\)
\(L(1)\) \(\approx\) \(0.6134210820 - 0.08694026314i\)
\(L(1)\) \(\approx\) \(0.6134210820 - 0.08694026314i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.967 - 0.251i)T \)
5 \( 1 + (-0.611 + 0.791i)T \)
11 \( 1 + (-0.316 - 0.948i)T \)
13 \( 1 + (0.493 - 0.869i)T \)
17 \( 1 + (0.717 - 0.696i)T \)
19 \( 1 + (-0.629 + 0.777i)T \)
23 \( 1 + (-0.599 - 0.800i)T \)
29 \( 1 + (-0.640 + 0.767i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.420 - 0.907i)T \)
43 \( 1 + (-0.178 + 0.983i)T \)
47 \( 1 + (0.862 - 0.506i)T \)
53 \( 1 + (0.961 - 0.273i)T \)
59 \( 1 + (0.337 + 0.941i)T \)
61 \( 1 + (-0.119 + 0.992i)T \)
67 \( 1 + (-0.998 + 0.0523i)T \)
71 \( 1 + (-0.767 + 0.640i)T \)
73 \( 1 + (0.930 + 0.365i)T \)
79 \( 1 + (0.258 + 0.965i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (0.00747 - 0.999i)T \)
97 \( 1 + (-0.453 + 0.891i)T \)
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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

−17.71669139517817426908137719145, −17.14158179452079360782971292123, −16.73081216679821206449743459030, −15.938255162899691700449629468769, −15.33236358648229158296278236734, −15.059215118808344915528300727434, −13.97100760560089223586066878250, −13.24952453976287775170094171887, −12.355018940483491607222752540938, −11.88078017955973242329865157665, −11.26184255706508297344136966800, −10.45002783671474636433111103851, −9.7602924908327182249080073972, −9.16506147631004303576454970305, −8.50446499825196069259970273498, −7.8786389091712794709797461252, −7.35274518879717391029943911550, −6.52246877077365669981841620492, −5.83354366666328036138088244283, −4.95422453849151270531358447073, −4.2597947854501419604457897011, −3.43986832848522966501016042289, −2.258018360022053762040883482451, −1.6722779850311585143707536908, −0.81223305289665768562029963555, 0.356290508401694208155908277494, 1.13263613315720914000974409058, 2.347137665119336787096131971048, 2.93005950609787598909370376785, 3.542603256692156086438417330092, 4.27687513163330354869637927968, 5.75648582487507354609337554024, 6.010137135385101898840624283020, 7.059927435108382055150499445030, 7.56053746547283107560889159218, 8.35959870211048612853327385390, 8.57202616843112220875423526710, 9.74743388398901135344162403179, 10.3818534078432093131805694781, 10.75947566676730568315066900317, 11.46310525339538203532601274157, 12.06451513166665582160962469392, 12.749165128713410230243770857979, 13.59378419338139864284462533982, 14.49068994050378423429653553647, 15.012740796365470168997189344649, 15.80912137699086119732475128101, 16.3288915971117741513501675292, 16.768276823631059237120345808927, 17.839113316827896368260968123025

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