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Properties

Label 1-6e3-216.157-r0-0-0
Degree $1$
Conductor $216$
Sign $-0.727 - 0.686i$
Analytic cond. $1.00309$
Root an. cond. $1.00309$
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  + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)7-s + (−0.766 − 0.642i)11-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)29-s + (−0.939 − 0.342i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (0.173 + 0.984i)41-s + (−0.766 − 0.642i)43-s + (−0.939 + 0.342i)47-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)7-s + (−0.766 − 0.642i)11-s + (−0.173 − 0.984i)13-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)23-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)29-s + (−0.939 − 0.342i)31-s + (0.5 − 0.866i)35-s + (0.5 + 0.866i)37-s + (0.173 + 0.984i)41-s + (−0.766 − 0.642i)43-s + (−0.939 + 0.342i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.727 - 0.686i$
Analytic conductor: \(1.00309\)
Root analytic conductor: \(1.00309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (0:\ ),\ -0.727 - 0.686i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09422033183 - 0.2371667773i\)
\(L(\frac12)\) \(\approx\) \(0.09422033183 - 0.2371667773i\)
\(L(1)\) \(\approx\) \(0.6103486696 - 0.04193698862i\)
\(L(1)\) \(\approx\) \(0.6103486696 - 0.04193698862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.939 - 0.342i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (-0.939 - 0.342i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−26.74033347984120518570733367878, −26.20595310655504058230488815889, −25.087086194912618248039138462668, −23.93536713136286775238278816058, −23.37148232688479194674760862333, −22.412084162157979858039678297438, −21.23782181580700644331663750441, −20.211212184835010614571202174924, −19.53394413782906052525090067707, −18.60883911849420602074762696698, −17.31464830381617793052146830638, −16.27793613040103097702861698320, −15.76132896740918773481221597018, −14.52435289378225608595852143129, −13.21301953110911617171583977288, −12.51861691834499286622940126975, −11.51640394010982755649338599196, −10.2007761932626982755993986266, −9.29495836749853104636681305604, −8.04734195801438901737435136107, −7.138590359219185693989003814946, −5.8404705947770560238118469915, −4.43379228966387336245876542683, −3.59487102266392707172237444832, −1.874128095346685694068944653144, 0.17895339569723269833167895635, 2.72172279793092340465433608656, 3.34627101102771266446788011582, 4.96090426558877433928514937990, 6.21663194195740069347446820321, 7.28918991285520900219158507126, 8.28608128732095192239348848737, 9.57049845685987228634523839804, 10.64972138400499260253727495095, 11.55788262334474039722144903215, 12.70540354066343670379895165335, 13.5995520869268042014705021253, 14.9190449482866916108070127241, 15.77199777922364134754357763184, 16.36444359591367247088902798371, 18.07064190187734875583399733697, 18.54350860020418785809416196275, 19.72002252494901267923844060114, 20.30500514362544306119073921220, 21.97467667638953752956028531745, 22.33488844444855328983073989994, 23.40419980533560391341393150768, 24.27880782117551925728237646796, 25.46644872232477104758483172664, 26.29289927724983319111304629495

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