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Properties

Label 1-432-432.349-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: $\chi_{432} (349, \cdot )$
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

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

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