Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} $
Sign $0.629 - 0.777i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.88 + 8.45i)5-s + (68.3 + 118. i)7-s + (−326. − 565. i)11-s + (−125. + 216. i)13-s − 249.·17-s + 1.75e3·19-s + (827. − 1.43e3i)23-s + (1.51e3 + 2.62e3i)25-s + (2.12e3 + 3.67e3i)29-s + (4.49e3 − 7.78e3i)31-s − 1.33e3·35-s − 6.00e3·37-s + (−5.37e3 + 9.30e3i)41-s + (5.02e3 + 8.70e3i)43-s + (−1.17e4 − 2.03e4i)47-s + ⋯
L(s)  = 1  + (−0.0873 + 0.151i)5-s + (0.527 + 0.912i)7-s + (−0.813 − 1.40i)11-s + (−0.205 + 0.356i)13-s − 0.209·17-s + 1.11·19-s + (0.326 − 0.564i)23-s + (0.484 + 0.839i)25-s + (0.468 + 0.812i)29-s + (0.839 − 1.45i)31-s − 0.184·35-s − 0.720·37-s + (−0.499 + 0.864i)41-s + (0.414 + 0.717i)43-s + (−0.775 − 1.34i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(432\)    =    \(2^{4} \cdot 3^{3}\)
\( \varepsilon \)  =  $0.629 - 0.777i$
motivic weight  =  \(5\)
character  :  $\chi_{432} (145, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 432,\ (\ :5/2),\ 0.629 - 0.777i)\)
\(L(3)\)  \(\approx\)  \(1.954177011\)
\(L(\frac12)\)  \(\approx\)  \(1.954177011\)
\(L(\frac{7}{2})\)   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 \( 1 + (4.88 - 8.45i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (-68.3 - 118. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (326. + 565. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (125. - 216. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 249.T + 1.41e6T^{2} \)
19 \( 1 - 1.75e3T + 2.47e6T^{2} \)
23 \( 1 + (-827. + 1.43e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (-2.12e3 - 3.67e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-4.49e3 + 7.78e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 6.00e3T + 6.93e7T^{2} \)
41 \( 1 + (5.37e3 - 9.30e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-5.02e3 - 8.70e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.17e4 + 2.03e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + 9.41e3T + 4.18e8T^{2} \)
59 \( 1 + (2.20e4 - 3.82e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.12e4 - 1.94e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.80e4 - 3.11e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 7.85e4T + 1.80e9T^{2} \)
73 \( 1 - 6.13e4T + 2.07e9T^{2} \)
79 \( 1 + (-1.37e4 - 2.38e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.24e4 - 5.61e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 3.46e4T + 5.58e9T^{2} \)
97 \( 1 + (8.05e3 + 1.39e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−10.62895986277527252150731905831, −9.466056820909434725718605892384, −8.578497056097429746619707113572, −7.914474411143967933288640486110, −6.70084766826754243319451522568, −5.60509611077855188470790078476, −4.91301705343791523184785251748, −3.33929166003255322496386003587, −2.44057710554425747284213834849, −0.926963819788458133038375863659, 0.58906877650881365701555779585, 1.85480906707752265098882107183, 3.21244854621139626160118648576, 4.60924911749265155399141310838, 5.09844135080443075608856722447, 6.68414936244229300132673053178, 7.52262402263250639004297359110, 8.161276509778454153719532003038, 9.491975084788547382161763809972, 10.25655502501303624327616440012

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