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Properties

Label 1-6e3-216.101-r1-0-0
Degree $1$
Conductor $216$
Sign $0.286 - 0.957i$
Analytic cond. $23.2124$
Root an. cond. $23.2124$
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  + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)7-s + (−0.939 − 0.342i)11-s + (−0.766 − 0.642i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (0.766 − 0.642i)29-s + (0.173 − 0.984i)31-s + (−0.5 − 0.866i)35-s + (0.5 − 0.866i)37-s + (−0.766 − 0.642i)41-s + (0.939 + 0.342i)43-s + (−0.173 − 0.984i)47-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)7-s + (−0.939 − 0.342i)11-s + (−0.766 − 0.642i)13-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (0.766 − 0.642i)29-s + (0.173 − 0.984i)31-s + (−0.5 − 0.866i)35-s + (0.5 − 0.866i)37-s + (−0.766 − 0.642i)41-s + (0.939 + 0.342i)43-s + (−0.173 − 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.286 - 0.957i$
Analytic conductor: \(23.2124\)
Root analytic conductor: \(23.2124\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 216,\ (1:\ ),\ 0.286 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7115122058 - 0.5297012112i\)
\(L(\frac12)\) \(\approx\) \(0.7115122058 - 0.5297012112i\)
\(L(1)\) \(\approx\) \(0.8098263692 + 0.007599134033i\)
\(L(1)\) \(\approx\) \(0.8098263692 + 0.007599134033i\)

Euler product

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

−26.61324615188648957941474658738, −25.86226188422276634078212959518, −24.352050963539092513885046511172, −23.74981642611699348954991296394, −23.10083763715311150868001216549, −21.85647759621027438449509886066, −20.74538404743041905617664664063, −19.95295398720761237871101804178, −19.20075249591318887677585850680, −17.98971133111020797327900053854, −16.901611149568144033078876228947, −16.13654053241246339973968248454, −15.08433066213800218065510378749, −14.089748983241077371251938377264, −12.89713306942594972587207941361, −12.04981593576770666201403433492, −10.88918765278976187389302561613, −10.0242557889045776030681191698, −8.5706824707800043008796478142, −7.6377616259794256373263769862, −6.80747265795337793332323336680, −4.97953665145384327972613197141, −4.27338271735717545101961462324, −2.88683474674098992425861463691, −1.083475968850367685515651569403, 0.34789498564892311157838332082, 2.435218313907227419991784213780, 3.400077409366433848935177915670, 4.95318977212905844796791056102, 5.874410486107810957660643424644, 7.5535282837688542448117537920, 8.01835196201406079203952539417, 9.432420712534912562960444298722, 10.55777265508963406386731212627, 11.75845757343831505330941739149, 12.287843582326980397401457374938, 13.66691048031388580852448427580, 14.8870352342504706847536690050, 15.55756513051328165762282540082, 16.39695646060765997369536000369, 17.88870733390780617203282587792, 18.6215335393003128595146836955, 19.427847407072707755373224000636, 20.54130562117927245629689145438, 21.50896825143965529716769669050, 22.54399446134798160825539464960, 23.27538635402721649287463729289, 24.36899776302275159153007410205, 25.12889269486362707975759291423, 26.28905889226535778069957686350

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