Properties

Label 2-432-9.2-c6-0-16
Degree $2$
Conductor $432$
Sign $0.968 - 0.250i$
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  + (39.5 − 22.8i)5-s + (245. − 424. i)7-s + (−873. − 504. i)11-s + (466. + 808. i)13-s + 8.09e3i·17-s + 7.72e3·19-s + (−1.18e4 + 6.84e3i)23-s + (−6.76e3 + 1.17e4i)25-s + (1.96e3 + 1.13e3i)29-s + (1.70e4 + 2.95e4i)31-s − 2.23e4i·35-s + 9.20e4·37-s + (3.10e4 − 1.79e4i)41-s + (−3.45e4 + 5.98e4i)43-s + (1.32e4 + 7.62e3i)47-s + ⋯
L(s)  = 1  + (0.316 − 0.182i)5-s + (0.714 − 1.23i)7-s + (−0.656 − 0.378i)11-s + (0.212 + 0.368i)13-s + 1.64i·17-s + 1.12·19-s + (−0.973 + 0.562i)23-s + (−0.433 + 0.750i)25-s + (0.0805 + 0.0465i)29-s + (0.572 + 0.992i)31-s − 0.522i·35-s + 1.81·37-s + (0.450 − 0.259i)41-s + (−0.434 + 0.753i)43-s + (0.127 + 0.0734i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(99.3833\)
Root analytic conductor: \(9.96912\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :3),\ 0.968 - 0.250i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.436867017\)
\(L(\frac12)\) \(\approx\) \(2.436867017\)
\(L(4)\) 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 \( 1 + (-39.5 + 22.8i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-245. + 424. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (873. + 504. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (-466. - 808. i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 - 8.09e3iT - 2.41e7T^{2} \)
19 \( 1 - 7.72e3T + 4.70e7T^{2} \)
23 \( 1 + (1.18e4 - 6.84e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-1.96e3 - 1.13e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-1.70e4 - 2.95e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 9.20e4T + 2.56e9T^{2} \)
41 \( 1 + (-3.10e4 + 1.79e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (3.45e4 - 5.98e4i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-1.32e4 - 7.62e3i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + 2.36e5iT - 2.21e10T^{2} \)
59 \( 1 + (2.21e5 - 1.28e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (1.99e4 - 3.45e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.60e5 - 2.77e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 4.04e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.93e5T + 1.51e11T^{2} \)
79 \( 1 + (-4.49e5 + 7.78e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-1.54e5 - 8.94e4i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 8.26e5iT - 4.96e11T^{2} \)
97 \( 1 + (3.17e5 - 5.50e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−10.29833803839978368538975468045, −9.404953017536188519003340886049, −8.104843859554356773623174533383, −7.69822128262796120034881397269, −6.41983558237399757176423055089, −5.43629872947506035332569931184, −4.35020289239260850299185169415, −3.41811040110826715284579175968, −1.79447442261122639244784105296, −0.937930720580063812574962990664, 0.63669993423865492256395380822, 2.20360545667999511302458041667, 2.80207146220647636336962681959, 4.52199598263062352957189262162, 5.40284236183346077254348011202, 6.17920891250915470579592329790, 7.57482782985247307813507164227, 8.188364008682470806005380891231, 9.366523215296130302426466828681, 9.941116931743226609779079381284

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