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Properties

Label 2-432-36.11-c1-0-2
Degree $2$
Conductor $432$
Sign $0.642 - 0.766i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)5-s + (1.5 + 0.866i)7-s + (1.5 − 2.59i)11-s + (2.5 + 4.33i)13-s + 6.92i·17-s − 3.46i·19-s + (4.5 + 7.79i)23-s + (−1 + 1.73i)25-s + (−1.5 − 0.866i)29-s + (4.5 − 2.59i)31-s − 3·35-s + 2·37-s + (4.5 − 2.59i)41-s + (−4.5 − 2.59i)43-s + (1.5 − 2.59i)47-s + ⋯
L(s)  = 1  + (−0.670 + 0.387i)5-s + (0.566 + 0.327i)7-s + (0.452 − 0.783i)11-s + (0.693 + 1.20i)13-s + 1.68i·17-s − 0.794i·19-s + (0.938 + 1.62i)23-s + (−0.200 + 0.346i)25-s + (−0.278 − 0.160i)29-s + (0.808 − 0.466i)31-s − 0.507·35-s + 0.328·37-s + (0.702 − 0.405i)41-s + (−0.686 − 0.396i)43-s + (0.218 − 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20330 + 0.561110i\)
\(L(\frac12)\) \(\approx\) \(1.20330 + 0.561110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.92iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.5 + 2.59i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-7.5 - 4.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.49480562415916876643654597770, −10.64836991281827918387494397530, −9.257852026377012874170797387807, −8.596117384397092222305506070525, −7.64869392854917781582324972832, −6.60817491719715258815622156760, −5.65777309537100298993665121022, −4.24936174981365544820299854820, −3.39269530903321715986331144385, −1.63224767827799364785894026218, 0.972153551011828083205733325427, 2.89311786311426452498601074402, 4.29061024415909365752001350641, 5.00712821129573869521797740142, 6.39714312918828389055270540308, 7.51997457001879806996451301490, 8.171149426208863069567925434265, 9.143191477207607612575391836339, 10.22979345755184869124366771898, 11.04460012093023587142421936429

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