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Properties

Label 2-432-3.2-c4-0-13
Degree $2$
Conductor $432$
Sign $-i$
Analytic cond. $44.6558$
Root an. cond. $6.68250$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 33i·5-s + 19·7-s + 123i·11-s + 302·13-s − 414i·17-s + 304·19-s + 300i·23-s − 464·25-s + 678i·29-s − 239·31-s + 627i·35-s + 740·37-s − 228i·41-s + 982·43-s + 2.16e3i·47-s + ⋯
L(s)  = 1  + 1.32i·5-s + 0.387·7-s + 1.01i·11-s + 1.78·13-s − 1.43i·17-s + 0.842·19-s + 0.567i·23-s − 0.742·25-s + 0.806i·29-s − 0.248·31-s + 0.511i·35-s + 0.540·37-s − 0.135i·41-s + 0.531·43-s + 0.980i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-i$
Analytic conductor: \(44.6558\)
Root analytic conductor: \(6.68250\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :2),\ -i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.287693114\)
\(L(\frac12)\) \(\approx\) \(2.287693114\)
\(L(3)\) 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 \( 1 - 33iT - 625T^{2} \)
7 \( 1 - 19T + 2.40e3T^{2} \)
11 \( 1 - 123iT - 1.46e4T^{2} \)
13 \( 1 - 302T + 2.85e4T^{2} \)
17 \( 1 + 414iT - 8.35e4T^{2} \)
19 \( 1 - 304T + 1.30e5T^{2} \)
23 \( 1 - 300iT - 2.79e5T^{2} \)
29 \( 1 - 678iT - 7.07e5T^{2} \)
31 \( 1 + 239T + 9.23e5T^{2} \)
37 \( 1 - 740T + 1.87e6T^{2} \)
41 \( 1 + 228iT - 2.82e6T^{2} \)
43 \( 1 - 982T + 3.41e6T^{2} \)
47 \( 1 - 2.16e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.59e3iT - 7.89e6T^{2} \)
59 \( 1 + 2.92e3iT - 1.21e7T^{2} \)
61 \( 1 + 316T + 1.38e7T^{2} \)
67 \( 1 + 4.62e3T + 2.01e7T^{2} \)
71 \( 1 - 1.81e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.03e3T + 2.83e7T^{2} \)
79 \( 1 - 1.04e4T + 3.89e7T^{2} \)
83 \( 1 - 1.26e4iT - 4.74e7T^{2} \)
89 \( 1 - 7.00e3iT - 6.27e7T^{2} \)
97 \( 1 + 6.51e3T + 8.85e7T^{2} \)
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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

−11.01291695289338665195092322068, −9.899310952965056480293515345853, −9.087360742199890235217641431341, −7.77244114411234898435382920482, −7.10318271440601160754626563773, −6.19787216911581109782498043429, −5.01502644700367161956130388076, −3.68561599692514857801255680692, −2.72057606706938499083571241025, −1.30723021924889872477913291184, 0.71613391535393847887768168932, 1.58674060797695215265115305654, 3.46903325266830113036909626391, 4.41178519547947471568620883523, 5.61916975788905037051490954415, 6.24307762093820969345085424808, 7.938443686123466830008390377042, 8.509852563490533973890280676430, 9.102404334340660728381761442422, 10.42268196153872591871060427051

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