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Properties

Label 2-432-36.11-c5-0-28
Degree $2$
Conductor $432$
Sign $-0.856 - 0.516i$
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  + (35.2 − 20.3i)5-s + (−165. − 95.6i)7-s + (104. − 181. i)11-s + (85.5 + 148. i)13-s + 182. i·17-s − 2.01e3i·19-s + (−841. − 1.45e3i)23-s + (−733. + 1.27e3i)25-s + (256. + 147. i)29-s + (2.89e3 − 1.67e3i)31-s − 7.78e3·35-s − 2.09e3·37-s + (−1.49e4 + 8.65e3i)41-s + (1.36e4 + 7.85e3i)43-s + (−1.35e4 + 2.34e4i)47-s + ⋯
L(s)  = 1  + (0.630 − 0.364i)5-s + (−1.27 − 0.737i)7-s + (0.260 − 0.451i)11-s + (0.140 + 0.243i)13-s + 0.153i·17-s − 1.27i·19-s + (−0.331 − 0.574i)23-s + (−0.234 + 0.406i)25-s + (0.0565 + 0.0326i)29-s + (0.541 − 0.312i)31-s − 1.07·35-s − 0.252·37-s + (−1.39 + 0.804i)41-s + (1.12 + 0.647i)43-s + (−0.894 + 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.856 - 0.516i$
Analytic conductor: \(69.2858\)
Root analytic conductor: \(8.32380\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :5/2),\ -0.856 - 0.516i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1905724739\)
\(L(\frac12)\) \(\approx\) \(0.1905724739\)
\(L(\frac{7}{2})\) 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 \( 1 + (-35.2 + 20.3i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (165. + 95.6i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (-104. + 181. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (-85.5 - 148. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 182. iT - 1.41e6T^{2} \)
19 \( 1 + 2.01e3iT - 2.47e6T^{2} \)
23 \( 1 + (841. + 1.45e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-256. - 147. i)T + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-2.89e3 + 1.67e3i)T + (1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + 2.09e3T + 6.93e7T^{2} \)
41 \( 1 + (1.49e4 - 8.65e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-1.36e4 - 7.85e3i)T + (7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (1.35e4 - 2.34e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 3.65e4iT - 4.18e8T^{2} \)
59 \( 1 + (1.28e4 + 2.22e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.75e4 - 3.04e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (2.09e4 - 1.21e4i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 3.08e4T + 1.80e9T^{2} \)
73 \( 1 - 5.50e4T + 2.07e9T^{2} \)
79 \( 1 + (4.19e4 + 2.42e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.82e4 + 6.62e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 - 1.42e5iT - 5.58e9T^{2} \)
97 \( 1 + (3.61e4 - 6.26e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−9.676411654982098450822814449732, −9.181395575770617512526524103474, −8.014964722858623416582210518575, −6.72309242452777590205593698515, −6.26466811240157251949192570531, −4.96874901781111391313684601236, −3.80568636986473741768804823268, −2.73091077583274676174018060465, −1.20283757559910991469928869440, −0.04582441980526964593393728074, 1.73936413675581488478451126635, 2.84547765460554814572917468029, 3.86089636482622343403922680448, 5.45590141788312820900551542167, 6.16276723849165992398001824964, 6.96817447083354349537205980127, 8.220745802122997753150525096967, 9.289287801406467928245229268809, 9.919366597904549425703742187442, 10.59870403644811698937842413529

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