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Properties

Label 1-432-432.157-r0-0-0
Degree $1$
Conductor $432$
Sign $-0.244 - 0.969i$
Analytic cond. $2.00619$
Root an. cond. $2.00619$
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  + (−0.642 − 0.766i)5-s + (0.939 − 0.342i)7-s + (0.642 − 0.766i)11-s + (−0.984 + 0.173i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.939 + 0.342i)47-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (0.939 − 0.342i)7-s + (0.642 − 0.766i)11-s + (−0.984 + 0.173i)13-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (0.984 + 0.173i)29-s + (−0.939 − 0.342i)31-s + (−0.866 − 0.5i)35-s + (−0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (0.642 − 0.766i)43-s + (−0.939 + 0.342i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.244 - 0.969i$
Analytic conductor: \(2.00619\)
Root analytic conductor: \(2.00619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (0:\ ),\ -0.244 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6463781913 - 0.8298112542i\)
\(L(\frac12)\) \(\approx\) \(0.6463781913 - 0.8298112542i\)
\(L(1)\) \(\approx\) \(0.9067143276 - 0.3322842621i\)
\(L(1)\) \(\approx\) \(0.9067143276 - 0.3322842621i\)

Euler product

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

−24.40753473224295835380373794561, −23.42609507084742322002228686274, −22.714778754228897255205509504283, −21.81554587975566087932031840700, −21.08032057314601909050036900598, −19.80606057559467982009675818078, −19.38841016432257611260487160645, −18.240148692123839138207005839117, −17.55520462076721298707690295656, −16.68219955799633981472045465746, −15.215634154645746696618144181434, −14.91651332625682379423195769099, −14.19552353851069911956791004396, −12.642760319394469677877138770662, −12.02888855870909711174539816222, −11.01089779120195298250029906649, −10.32399308974615137727774620495, −9.01922877855396105597916039010, −8.06133354077570644776349994128, −7.18987173535671221456605839566, −6.28034127523154790274334984390, −4.847883325356986332864764093048, −4.09929369360335605308899210943, −2.73383231884504591336341459630, −1.683279997202114372828874860750, 0.60813931132744238353925019947, 1.946878669699214766640249228985, 3.46230214035425405679359173526, 4.61606799865373345673591724259, 5.14108275251835911042331209451, 6.738032041098803915264779532594, 7.60993239718546848012436465943, 8.63591663160930656253873183156, 9.250460298201771659503065161795, 10.744931627727995913924618788639, 11.492453045547847262808185647530, 12.23727341219182865049540689537, 13.34306553250428018817721454157, 14.25401717390677893297315685647, 15.12681399174626753377082286113, 16.09533568507505823118844840287, 17.04839535260748794288345626489, 17.51593536703072884113191422451, 18.91881524893739561997586051246, 19.59933705858437415948469910119, 20.415590643816441252548530399110, 21.23294736455088874219039447659, 22.09939608136755233763850630595, 23.175479817733857832939379975277, 24.13482823997650363439351901419

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