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Properties

Label 2-432-9.5-c6-0-18
Degree $2$
Conductor $432$
Sign $0.919 - 0.394i$
Analytic cond. $99.3833$
Root an. cond. $9.96912$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (157. + 90.9i)5-s + (−83.3 − 144. i)7-s + (1.48e3 − 859. i)11-s + (−279. + 483. i)13-s + 5.31e3i·17-s + 1.40e3·19-s + (3.65e3 + 2.11e3i)23-s + (8.73e3 + 1.51e4i)25-s + (2.79e3 − 1.61e3i)29-s + (2.22e4 − 3.85e4i)31-s − 3.03e4i·35-s + 6.45e3·37-s + (5.32e4 + 3.07e4i)41-s + (1.13e4 + 1.97e4i)43-s + (−3.89e4 + 2.24e4i)47-s + ⋯
L(s)  = 1  + (1.26 + 0.727i)5-s + (−0.243 − 0.420i)7-s + (1.11 − 0.645i)11-s + (−0.127 + 0.220i)13-s + 1.08i·17-s + 0.205·19-s + (0.300 + 0.173i)23-s + (0.559 + 0.968i)25-s + (0.114 − 0.0662i)29-s + (0.746 − 1.29i)31-s − 0.707i·35-s + 0.127·37-s + (0.773 + 0.446i)41-s + (0.143 + 0.247i)43-s + (−0.375 + 0.216i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.919 - 0.394i$
Analytic conductor: \(99.3833\)
Root analytic conductor: \(9.96912\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :3),\ 0.919 - 0.394i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.169214510\)
\(L(\frac12)\) \(\approx\) \(3.169214510\)
\(L(4)\) 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 \( 1 + (-157. - 90.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (83.3 + 144. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-1.48e3 + 859. i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (279. - 483. i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 - 5.31e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.40e3T + 4.70e7T^{2} \)
23 \( 1 + (-3.65e3 - 2.11e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-2.79e3 + 1.61e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (-2.22e4 + 3.85e4i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 6.45e3T + 2.56e9T^{2} \)
41 \( 1 + (-5.32e4 - 3.07e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (-1.13e4 - 1.97e4i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (3.89e4 - 2.24e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + 7.23e4iT - 2.21e10T^{2} \)
59 \( 1 + (1.75e5 + 1.01e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-1.81e5 - 3.13e5i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (2.44e5 - 4.23e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 1.52e5iT - 1.28e11T^{2} \)
73 \( 1 - 1.78e4T + 1.51e11T^{2} \)
79 \( 1 + (1.46e5 + 2.52e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-3.56e5 + 2.05e5i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 + 7.58e5iT - 4.96e11T^{2} \)
97 \( 1 + (4.77e5 + 8.27e5i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−10.12114030153729747960525666261, −9.514373072830510711430054028027, −8.538164868793212545478049297708, −7.25690074006013658622625638261, −6.28831645653943998791418382215, −5.88304728280300293786329972175, −4.30428565350547031279483527622, −3.22437055088211673415759639657, −2.05402710593740219711429742020, −0.960247327134622628564553944868, 0.846027147199219648797957577801, 1.82764782321997478165956395491, 2.95007074379729462014530621552, 4.51404662256818399038285209283, 5.34067818727628343820252085833, 6.28546311647705752378324973574, 7.16950620836724798661770188390, 8.576699795361406152607385646145, 9.388611408546653465782931029336, 9.730377453578414719943974779299

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