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Properties

Label 2-432-36.7-c6-0-17
Degree $2$
Conductor $432$
Sign $0.510 + 0.859i$
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  + (39.8 + 69.0i)5-s + (−343. − 198. i)7-s + (−201. − 116. i)11-s + (−109. − 188. i)13-s + 3.48e3·17-s + 9.75e3i·19-s + (−7.74e3 + 4.47e3i)23-s + (4.63e3 − 8.02e3i)25-s + (−5.17e3 + 8.95e3i)29-s + (−2.39e4 + 1.38e4i)31-s − 3.16e4i·35-s + 5.20e4·37-s + (−5.91e4 − 1.02e5i)41-s + (−3.11e4 − 1.79e4i)43-s + (3.37e4 + 1.94e4i)47-s + ⋯
L(s)  = 1  + (0.318 + 0.552i)5-s + (−1.00 − 0.578i)7-s + (−0.151 − 0.0875i)11-s + (−0.0496 − 0.0859i)13-s + 0.708·17-s + 1.42i·19-s + (−0.636 + 0.367i)23-s + (0.296 − 0.513i)25-s + (−0.212 + 0.367i)29-s + (−0.804 + 0.464i)31-s − 0.737i·35-s + 1.02·37-s + (−0.858 − 1.48i)41-s + (−0.391 − 0.225i)43-s + (0.324 + 0.187i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.510 + 0.859i$
Analytic conductor: \(99.3833\)
Root analytic conductor: \(9.96912\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :3),\ 0.510 + 0.859i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.382254354\)
\(L(\frac12)\) \(\approx\) \(1.382254354\)
\(L(4)\) 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 \( 1 + (-39.8 - 69.0i)T + (-7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (343. + 198. i)T + (5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (201. + 116. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (109. + 188. i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 - 3.48e3T + 2.41e7T^{2} \)
19 \( 1 - 9.75e3iT - 4.70e7T^{2} \)
23 \( 1 + (7.74e3 - 4.47e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (5.17e3 - 8.95e3i)T + (-2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (2.39e4 - 1.38e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 5.20e4T + 2.56e9T^{2} \)
41 \( 1 + (5.91e4 + 1.02e5i)T + (-2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (3.11e4 + 1.79e4i)T + (3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (-3.37e4 - 1.94e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + 8.65e3T + 2.21e10T^{2} \)
59 \( 1 + (-2.84e5 + 1.64e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-1.10e4 + 1.92e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (3.91e5 - 2.25e5i)T + (4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 5.72e5iT - 1.28e11T^{2} \)
73 \( 1 + 8.95e4T + 1.51e11T^{2} \)
79 \( 1 + (-1.69e5 - 9.80e4i)T + (1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-2.05e5 - 1.18e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 1.31e6T + 4.96e11T^{2} \)
97 \( 1 + (-5.60e5 + 9.71e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−10.22221565083947175503470882428, −9.303171218800752464753226441633, −8.073449958524985592777183638804, −7.17034921062705258410009314934, −6.28749180659187356919593415474, −5.44356024001762489418347879695, −3.88058977206502450962861135242, −3.16965646042376783286805677456, −1.82142586444095653457862103318, −0.37499789391837246859550186713, 0.839535146815802537519211083723, 2.27425282710534713209170510362, 3.28577614221642349071978159298, 4.63313361927885535391321940094, 5.62482113123739832098523430068, 6.46523836198113934898585236420, 7.54493592819583077814526368089, 8.699889804823559978802945687473, 9.419433331274283850593235381320, 10.06305312491421676383633908730

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