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Properties

Label 1-6012-6012.11-r0-0-0
Degree $1$
Conductor $6012$
Sign $0.376 + 0.926i$
Analytic cond. $27.9196$
Root an. cond. $27.9196$
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  + (0.999 + 0.0126i)5-s + (0.904 + 0.427i)7-s + (−0.768 + 0.639i)11-s + (−0.267 − 0.963i)13-s + (−0.929 + 0.369i)17-s + (−0.206 + 0.978i)19-s + (0.979 + 0.200i)23-s + (0.999 + 0.0252i)25-s + (−0.979 + 0.200i)29-s + (0.996 + 0.0882i)31-s + (0.898 + 0.438i)35-s + (−0.243 + 0.969i)37-s + (0.340 + 0.940i)41-s + (0.543 − 0.839i)43-s + (0.909 − 0.415i)47-s + ⋯
L(s)  = 1  + (0.999 + 0.0126i)5-s + (0.904 + 0.427i)7-s + (−0.768 + 0.639i)11-s + (−0.267 − 0.963i)13-s + (−0.929 + 0.369i)17-s + (−0.206 + 0.978i)19-s + (0.979 + 0.200i)23-s + (0.999 + 0.0252i)25-s + (−0.979 + 0.200i)29-s + (0.996 + 0.0882i)31-s + (0.898 + 0.438i)35-s + (−0.243 + 0.969i)37-s + (0.340 + 0.940i)41-s + (0.543 − 0.839i)43-s + (0.909 − 0.415i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(27.9196\)
Root analytic conductor: \(27.9196\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6012} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6012,\ (0:\ ),\ 0.376 + 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.855915327 + 1.248478115i\)
\(L(\frac12)\) \(\approx\) \(1.855915327 + 1.248478115i\)
\(L(1)\) \(\approx\) \(1.301689339 + 0.2290515797i\)
\(L(1)\) \(\approx\) \(1.301689339 + 0.2290515797i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.999 + 0.0126i)T \)
7 \( 1 + (0.904 + 0.427i)T \)
11 \( 1 + (-0.768 + 0.639i)T \)
13 \( 1 + (-0.267 - 0.963i)T \)
17 \( 1 + (-0.929 + 0.369i)T \)
19 \( 1 + (-0.206 + 0.978i)T \)
23 \( 1 + (0.979 + 0.200i)T \)
29 \( 1 + (-0.979 + 0.200i)T \)
31 \( 1 + (0.996 + 0.0882i)T \)
37 \( 1 + (-0.243 + 0.969i)T \)
41 \( 1 + (0.340 + 0.940i)T \)
43 \( 1 + (0.543 - 0.839i)T \)
47 \( 1 + (0.909 - 0.415i)T \)
53 \( 1 + (0.752 - 0.658i)T \)
59 \( 1 + (-0.784 + 0.619i)T \)
61 \( 1 + (0.304 - 0.952i)T \)
67 \( 1 + (-0.510 - 0.859i)T \)
71 \( 1 + (-0.843 - 0.537i)T \)
73 \( 1 + (0.997 + 0.0756i)T \)
79 \( 1 + (0.682 + 0.731i)T \)
83 \( 1 + (0.875 - 0.483i)T \)
89 \( 1 + (-0.988 + 0.150i)T \)
97 \( 1 + (-0.996 + 0.0882i)T \)
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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

−17.75156948493563584100821929200, −16.96203479106078285781170478699, −16.47380383927081425811774434433, −15.56959189732803814332180831971, −14.93798568028535206348825389781, −14.13276406913864274203624844670, −13.67526470612845172772293110265, −13.233710561326140277764096786489, −12.41034450721263563387588109241, −11.406851855731809425722867504182, −10.93711261400036961877631832704, −10.510241515199802111880807320, −9.41693401467842848595434559825, −9.0300438772766717048671803787, −8.35460790024093707252867223485, −7.3134340160072065689885576164, −6.951553626974989763598658985786, −5.99570513283371924143908405141, −5.33980366150751508826688051867, −4.643702353592939806727907918423, −4.09933459793961049186319808143, −2.70996265206357292687177258979, −2.40295395955106373648417801365, −1.4827935249296935451100799318, −0.56894740097552924481865269028, 1.037220932987339684343034417352, 1.90923165837930861797927342584, 2.39313668601936081957995799463, 3.19590838286252391593529275256, 4.35499254750675711139423507649, 5.07004398800738204285946722269, 5.49414711208285274638056662540, 6.24606969405091670103881039522, 7.09138663965120389583170204641, 7.8847211261820012971863261676, 8.437006913552842210347664128528, 9.21253673126963020225640504509, 9.91154066320847149552809228988, 10.632250642445499041847336385810, 10.96134029191602168947607603740, 12.082261376575867403687617574911, 12.58155619276729651542991957895, 13.31997723903750512237063929295, 13.764655746112599722929118005154, 14.84515322764645344908191761116, 15.01629656354763338763447101814, 15.67524376298736687029821624032, 16.82389444873171321658164856927, 17.22550875905350924458121427559, 17.87120449989378166002037530105

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