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Properties

Label 2-432-9.7-c5-0-16
Degree $2$
Conductor $432$
Sign $0.687 - 0.726i$
Analytic cond. $69.2858$
Root an. cond. $8.32380$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (39.2 + 67.9i)5-s + (110. − 191. i)7-s + (−115. + 199. i)11-s + (385. + 668. i)13-s + 769.·17-s + 383.·19-s + (−193. − 334. i)23-s + (−1.51e3 + 2.62e3i)25-s + (−394. + 683. i)29-s + (1.60e3 + 2.78e3i)31-s + 1.73e4·35-s + 2.46e3·37-s + (−4.62e3 − 8.00e3i)41-s + (5.31e3 − 9.20e3i)43-s + (−976. + 1.69e3i)47-s + ⋯
L(s)  = 1  + (0.701 + 1.21i)5-s + (0.852 − 1.47i)7-s + (−0.286 + 0.496i)11-s + (0.633 + 1.09i)13-s + 0.646·17-s + 0.243·19-s + (−0.0761 − 0.131i)23-s + (−0.485 + 0.841i)25-s + (−0.0871 + 0.151i)29-s + (0.300 + 0.521i)31-s + 2.39·35-s + 0.296·37-s + (−0.429 − 0.743i)41-s + (0.438 − 0.759i)43-s + (−0.0644 + 0.111i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.687 - 0.726i$
Analytic conductor: \(69.2858\)
Root analytic conductor: \(8.32380\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :5/2),\ 0.687 - 0.726i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.904193892\)
\(L(\frac12)\) \(\approx\) \(2.904193892\)
\(L(\frac{7}{2})\) 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 \( 1 + (-39.2 - 67.9i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-110. + 191. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (115. - 199. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-385. - 668. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 769.T + 1.41e6T^{2} \)
19 \( 1 - 383.T + 2.47e6T^{2} \)
23 \( 1 + (193. + 334. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (394. - 683. i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (-1.60e3 - 2.78e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 2.46e3T + 6.93e7T^{2} \)
41 \( 1 + (4.62e3 + 8.00e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-5.31e3 + 9.20e3i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (976. - 1.69e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 - 3.25e4T + 4.18e8T^{2} \)
59 \( 1 + (-1.19e4 - 2.06e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.88e4 - 3.25e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.15e4 + 1.99e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.60e4T + 1.80e9T^{2} \)
73 \( 1 - 6.51e4T + 2.07e9T^{2} \)
79 \( 1 + (3.54e4 - 6.13e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.76e4 - 4.78e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 1.05e4T + 5.58e9T^{2} \)
97 \( 1 + (-4.14e4 + 7.17e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−10.46439440860496512686744582457, −9.904866491201269732822897522908, −8.616088410771564432903415649781, −7.33387412514136932020890166619, −7.01063541678174098225572093235, −5.84260214372578682742685448798, −4.54223309721499718662224313658, −3.57768335753677543583466105826, −2.19389548430297005113077299167, −1.12740245461596870478985412935, 0.799715271963220810157954565059, 1.83173223063805881320396583616, 3.04278717962441169952691306322, 4.72507438870421166561144032576, 5.57815174729452483168412642674, 5.92806946163499283730790830126, 7.88626450838342323794420193226, 8.446295960437982884442062473803, 9.162772498078115576658138290037, 10.08746057400290611007436588791

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