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Properties

Label 1-432-432.77-r1-0-0
Degree $1$
Conductor $432$
Sign $0.696 + 0.717i$
Analytic cond. $46.4248$
Root an. cond. $46.4248$
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  + (0.342 − 0.939i)5-s + (−0.173 + 0.984i)7-s + (−0.342 − 0.939i)11-s + (0.642 + 0.766i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)35-s + (−0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.173 + 0.984i)47-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)5-s + (−0.173 + 0.984i)7-s + (−0.342 − 0.939i)11-s + (0.642 + 0.766i)13-s + (0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.642 − 0.766i)29-s + (0.173 + 0.984i)31-s + (0.866 + 0.5i)35-s + (−0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.173 + 0.984i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.696 + 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.696 + 0.717i$
Analytic conductor: \(46.4248\)
Root analytic conductor: \(46.4248\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (1:\ ),\ 0.696 + 0.717i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696635383 + 0.7172685116i\)
\(L(\frac12)\) \(\approx\) \(1.696635383 + 0.7172685116i\)
\(L(1)\) \(\approx\) \(1.123860955 + 0.05937314639i\)
\(L(1)\) \(\approx\) \(1.123860955 + 0.05937314639i\)

Euler product

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

−23.4496281687130509308810133521, −22.956350912153800177025134816502, −22.37317660945569520521929184733, −21.00893875285894141015860632706, −20.53809114195712155862193240611, −19.4802471731763372663596577550, −18.47796969016334050040043772153, −17.84590374462129126753348570956, −16.94502452918801243446523060565, −15.94124779148121655075170839106, −14.91983473094445693240014928820, −14.20712258338365374719953065780, −13.290754735639607961078474122198, −12.43820388951483293551841055033, −11.09658282949327481368506929715, −10.36089700782947733704653144218, −9.82075374391666573474964479839, −8.337074912440005530072997952302, −7.29440656192943574340715030520, −6.65563945146793599388450091745, −5.48838347892824810838150894792, −4.22090288893847679266016372693, −3.19128977435857070984164600757, −2.09264202827234879630487093372, −0.56050749732884268500353117045, 1.058998411568333840894467577575, 2.19900953499705363603116097318, 3.49952421415537278344080081270, 4.74467572787830261363318528171, 5.7565874126788517374813212569, 6.38958283298294408447767970257, 8.09176700752719396240582652259, 8.72854962452372152745693490327, 9.459664754975811908973457709280, 10.70385027282671440261768034565, 11.72053458384528859016312374313, 12.5912896539105420328879314795, 13.3724811297764095676263791472, 14.26004288035947989554760604847, 15.56770945271928007732863591727, 16.0732136906264924891208534984, 17.07905058821896574184255905008, 17.870455072031605686669942033364, 19.149135734536007330116135101032, 19.36731091104087834417048599975, 21.07329718663918746590262339843, 21.19427586858252530820342487901, 22.05004169591526112951415295543, 23.4647038731866145664002336160, 23.95202003431738463953940798769

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