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Properties

Label 1-432-432.229-r0-0-0
Degree $1$
Conductor $432$
Sign $0.858 + 0.512i$
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  + (−0.984 + 0.173i)5-s + (−0.766 − 0.642i)7-s + (0.984 + 0.173i)11-s + (−0.342 + 0.939i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (0.342 + 0.939i)29-s + (0.766 − 0.642i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + (0.939 + 0.342i)41-s + (0.984 + 0.173i)43-s + (0.766 + 0.642i)47-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)5-s + (−0.766 − 0.642i)7-s + (0.984 + 0.173i)11-s + (−0.342 + 0.939i)13-s + (−0.5 − 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (0.342 + 0.939i)29-s + (0.766 − 0.642i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + (0.939 + 0.342i)41-s + (0.984 + 0.173i)43-s + (0.766 + 0.642i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.858 + 0.512i$
Analytic conductor: \(2.00619\)
Root analytic conductor: \(2.00619\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 432,\ (0:\ ),\ 0.858 + 0.512i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9324674418 + 0.2571361717i\)
\(L(\frac12)\) \(\approx\) \(0.9324674418 + 0.2571361717i\)
\(L(1)\) \(\approx\) \(0.8740201270 + 0.06314597952i\)
\(L(1)\) \(\approx\) \(0.8740201270 + 0.06314597952i\)

Euler product

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

−24.28430750052644022447871308108, −22.98644626347707281892351553351, −22.43412721120092238940635564675, −21.71556764796106652556138339604, −20.343653411906764055431156493669, −19.65753359753754345235328508522, −19.1536412933150053081374362409, −18.01641510126674313369756930852, −17.064043508876997939453778534385, −16.01820909349585645248684907431, −15.46496141277363781947160759263, −14.60470109299394542060909277600, −13.37305240851996039276219820581, −12.357630577849031292499514359554, −11.88645885664605753297766997087, −10.75512439460087101083868598098, −9.647880165518270787241690928623, −8.70587119924488291626669332535, −7.881246828358509598984444121257, −6.71273061567049310431260459678, −5.83026494341792044125768598973, −4.512318875100359253946850695311, −3.56078459701725427188741282302, −2.53765694683220441639890248698, −0.724493361386119403127348819142, 1.03536366010618082087568790729, 2.74770045565246058985958446269, 3.88744584116854374984896747306, 4.47512166004748823090368661652, 6.12229821959014519553608851182, 7.07610128548580060532250500595, 7.67103812869812614002480165482, 9.119510825966189932023869551898, 9.74134147919236782478197181650, 11.03910341589402597166056871759, 11.79364866796740665921290285365, 12.54862745489749743000436566922, 13.88047671861072964795977395328, 14.397397899836885880107893578022, 15.724590469007712987055938143128, 16.23523235683007493879969647145, 17.115155623718970649237151010026, 18.265376973597857144458516600865, 19.232378985824233950219766215930, 19.82917439790800107903662513113, 20.470828521006249316254774346148, 21.889332545691376771627982949152, 22.55814467135887104619188371713, 23.25784210504816496068842468998, 24.127371198957169996283137688211

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