Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} $
Sign $0.882 - 0.471i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (14.0 + 24.3i)5-s + (−75.7 + 131. i)7-s + (138. − 240. i)11-s + (−291. − 505. i)13-s + 1.61e3·17-s − 1.36e3·19-s + (−428. − 741. i)23-s + (1.16e3 − 2.02e3i)25-s + (4.26e3 − 7.39e3i)29-s + (1.46e3 + 2.54e3i)31-s − 4.26e3·35-s + 4.03e3·37-s + (9.44e3 + 1.63e4i)41-s + (−1.01e4 + 1.75e4i)43-s + (147. − 256. i)47-s + ⋯
L(s)  = 1  + (0.251 + 0.435i)5-s + (−0.583 + 1.01i)7-s + (0.346 − 0.599i)11-s + (−0.479 − 0.829i)13-s + 1.35·17-s − 0.869·19-s + (−0.168 − 0.292i)23-s + (0.373 − 0.646i)25-s + (0.942 − 1.63i)29-s + (0.274 + 0.475i)31-s − 0.587·35-s + 0.484·37-s + (0.877 + 1.52i)41-s + (−0.837 + 1.45i)43-s + (0.00976 − 0.0169i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.882 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(432\)    =    \(2^{4} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.882 - 0.471i$
motivic weight  =  \(5\)
character  :  $\chi_{432} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 432,\ (\ :5/2),\ 0.882 - 0.471i)\)
\(L(3)\)  \(\approx\)  \(2.021598792\)
\(L(\frac12)\)  \(\approx\)  \(2.021598792\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-14.0 - 24.3i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (75.7 - 131. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-138. + 240. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (291. + 505. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.61e3T + 1.41e6T^{2} \)
19 \( 1 + 1.36e3T + 2.47e6T^{2} \)
23 \( 1 + (428. + 741. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-4.26e3 + 7.39e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-1.46e3 - 2.54e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 4.03e3T + 6.93e7T^{2} \)
41 \( 1 + (-9.44e3 - 1.63e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (1.01e4 - 1.75e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (-147. + 256. i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 3.03e3T + 4.18e8T^{2} \)
59 \( 1 + (-8.61e3 - 1.49e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.28e4 - 2.22e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.31e4 + 2.27e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 7.66e4T + 1.80e9T^{2} \)
73 \( 1 - 1.49e3T + 2.07e9T^{2} \)
79 \( 1 + (-4.96e4 + 8.59e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.50e4 - 4.33e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 1.36e5T + 5.58e9T^{2} \)
97 \( 1 + (-3.33e4 + 5.77e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.20619045460245526240805162810, −9.708000482853899676519509199239, −8.536036976538362973416420667349, −7.81485985849324569756347367909, −6.30115376472958670395335624738, −6.01009593986190878796210481384, −4.67268451648028196785824038808, −3.16200520190747108147789072765, −2.50273607851355404842603884389, −0.78683999447502801480442605977, 0.70499618769518911154069849198, 1.86349394982845635563294180919, 3.43214368412301148143950989968, 4.38574599163132842078585104410, 5.44231386640853084306152614779, 6.75150863355054329430592952133, 7.28705950341043648003341953811, 8.542917662065977071172146989374, 9.531683870514860052290902055798, 10.12003001289081054199037936353

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