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Properties

Label 2-432-9.7-c7-0-23
Degree $2$
Conductor $432$
Sign $-0.436 + 0.899i$
Analytic cond. $134.950$
Root an. cond. $11.6168$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−156. − 270. i)5-s + (−626. + 1.08e3i)7-s + (992. − 1.71e3i)11-s + (6.03e3 + 1.04e4i)13-s + 5.45e3·17-s − 5.19e4·19-s + (−4.55e4 − 7.89e4i)23-s + (−9.81e3 + 1.69e4i)25-s + (−5.63e4 + 9.76e4i)29-s + (1.65e5 + 2.87e5i)31-s + 3.91e5·35-s + 3.13e5·37-s + (9.59e4 + 1.66e5i)41-s + (−8.43e4 + 1.46e5i)43-s + (−2.40e5 + 4.16e5i)47-s + ⋯
L(s)  = 1  + (−0.559 − 0.968i)5-s + (−0.689 + 1.19i)7-s + (0.224 − 0.389i)11-s + (0.762 + 1.32i)13-s + 0.269·17-s − 1.73·19-s + (−0.780 − 1.35i)23-s + (−0.125 + 0.217i)25-s + (−0.429 + 0.743i)29-s + (0.999 + 1.73i)31-s + 1.54·35-s + 1.01·37-s + (0.217 + 0.376i)41-s + (−0.161 + 0.280i)43-s + (−0.337 + 0.585i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.436 + 0.899i$
Analytic conductor: \(134.950\)
Root analytic conductor: \(11.6168\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :7/2),\ -0.436 + 0.899i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.5577211830\)
\(L(\frac12)\) \(\approx\) \(0.5577211830\)
\(L(\frac{9}{2})\) 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 \( 1 + (156. + 270. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (626. - 1.08e3i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-992. + 1.71e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + (-6.03e3 - 1.04e4i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 - 5.45e3T + 4.10e8T^{2} \)
19 \( 1 + 5.19e4T + 8.93e8T^{2} \)
23 \( 1 + (4.55e4 + 7.89e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (5.63e4 - 9.76e4i)T + (-8.62e9 - 1.49e10i)T^{2} \)
31 \( 1 + (-1.65e5 - 2.87e5i)T + (-1.37e10 + 2.38e10i)T^{2} \)
37 \( 1 - 3.13e5T + 9.49e10T^{2} \)
41 \( 1 + (-9.59e4 - 1.66e5i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (8.43e4 - 1.46e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + (2.40e5 - 4.16e5i)T + (-2.53e11 - 4.38e11i)T^{2} \)
53 \( 1 - 5.62e5T + 1.17e12T^{2} \)
59 \( 1 + (-5.23e5 - 9.06e5i)T + (-1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (-5.87e5 + 1.01e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (1.63e6 + 2.82e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 - 2.19e6T + 9.09e12T^{2} \)
73 \( 1 - 3.66e5T + 1.10e13T^{2} \)
79 \( 1 + (-1.01e6 + 1.76e6i)T + (-9.60e12 - 1.66e13i)T^{2} \)
83 \( 1 + (1.79e6 - 3.11e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + 5.04e6T + 4.42e13T^{2} \)
97 \( 1 + (-4.80e6 + 8.33e6i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−9.418365180567238663110140930089, −8.607494492192766228347141650007, −8.385467904409380166811782966068, −6.60597852511713116311355547202, −6.11297824000636656273238535466, −4.76593462109266685515177153308, −3.98704956076614617657595106852, −2.66854159402262513782351455967, −1.46335787558228347228346153943, −0.14380996549416970995783101723, 0.830249761765379436820007624526, 2.44273546578715173537666275722, 3.67380526048574063989446943432, 4.05838365290919309776230763907, 5.82822293226076805493325923706, 6.62564217005998315961010411642, 7.54994371593127213214583091076, 8.156619213920594166717600918960, 9.677732179018033003442336792247, 10.33183733891967414059524217847

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