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Properties

Label 2-432-12.11-c3-0-7
Degree $2$
Conductor $432$
Sign $0.5 - 0.866i$
Analytic cond. $25.4888$
Root an. cond. $5.04864$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.3i·5-s − 19.7i·7-s − 30.8·11-s + 72.3·13-s + 0.430i·17-s + 22.1i·19-s + 170.·23-s − 54.3·25-s + 269. i·29-s − 74.4i·31-s + 264.·35-s − 287.·37-s + 136. i·41-s + 407. i·43-s − 245.·47-s + ⋯
L(s)  = 1  + 1.19i·5-s − 1.06i·7-s − 0.844·11-s + 1.54·13-s + 0.00614i·17-s + 0.267i·19-s + 1.54·23-s − 0.434·25-s + 1.72i·29-s − 0.431i·31-s + 1.27·35-s − 1.27·37-s + 0.521i·41-s + 1.44i·43-s − 0.762·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(25.4888\)
Root analytic conductor: \(5.04864\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :3/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.821398984\)
\(L(\frac12)\) \(\approx\) \(1.821398984\)
\(L(\frac{5}{2})\) 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 \( 1 - 13.3iT - 125T^{2} \)
7 \( 1 + 19.7iT - 343T^{2} \)
11 \( 1 + 30.8T + 1.33e3T^{2} \)
13 \( 1 - 72.3T + 2.19e3T^{2} \)
17 \( 1 - 0.430iT - 4.91e3T^{2} \)
19 \( 1 - 22.1iT - 6.85e3T^{2} \)
23 \( 1 - 170.T + 1.21e4T^{2} \)
29 \( 1 - 269. iT - 2.43e4T^{2} \)
31 \( 1 + 74.4iT - 2.97e4T^{2} \)
37 \( 1 + 287.T + 5.06e4T^{2} \)
41 \( 1 - 136. iT - 6.89e4T^{2} \)
43 \( 1 - 407. iT - 7.95e4T^{2} \)
47 \( 1 + 245.T + 1.03e5T^{2} \)
53 \( 1 - 686. iT - 1.48e5T^{2} \)
59 \( 1 - 437.T + 2.05e5T^{2} \)
61 \( 1 - 437.T + 2.26e5T^{2} \)
67 \( 1 - 312. iT - 3.00e5T^{2} \)
71 \( 1 - 356.T + 3.57e5T^{2} \)
73 \( 1 - 434.T + 3.89e5T^{2} \)
79 \( 1 + 544. iT - 4.93e5T^{2} \)
83 \( 1 - 1.42e3T + 5.71e5T^{2} \)
89 \( 1 + 21.6iT - 7.04e5T^{2} \)
97 \( 1 - 273.T + 9.12e5T^{2} \)
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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

−10.69344978412292027976454736923, −10.44621605581503908981482522979, −9.082648171222983328083922898877, −7.991153887656540267129746061431, −7.08106157481519135457751010114, −6.42868904491133738703053107390, −5.12781150969048029937182887201, −3.72730151869941342984542205363, −2.94641020434827757739652459143, −1.18386756461899495653022573624, 0.69842932872474438857102378127, 2.14625649951397757742703708113, 3.59528133547501276180471731834, 5.02884692051678786947508877006, 5.54615845829771609931402528153, 6.75585627862294998995108857751, 8.268285137248121227487200601289, 8.628530589361231475681574597209, 9.452239599262307476977554599936, 10.66087079343432151446994885093

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