Properties

Label 2-432-9.4-c7-0-20
Degree $2$
Conductor $432$
Sign $0.926 - 0.377i$
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  + (145. − 251. i)5-s + (555. + 962. i)7-s + (−2.24e3 − 3.88e3i)11-s + (−1.21e3 + 2.11e3i)13-s − 1.59e4·17-s + 4.99e4·19-s + (−3.46e4 + 6.00e4i)23-s + (−3.16e3 − 5.47e3i)25-s + (−4.70e4 − 8.14e4i)29-s + (−9.96e3 + 1.72e4i)31-s + 3.23e5·35-s + 3.31e5·37-s + (1.21e5 − 2.09e5i)41-s + (4.15e5 + 7.20e5i)43-s + (8.00e4 + 1.38e5i)47-s + ⋯
L(s)  = 1  + (0.519 − 0.900i)5-s + (0.612 + 1.06i)7-s + (−0.508 − 0.880i)11-s + (−0.153 + 0.266i)13-s − 0.785·17-s + 1.67·19-s + (−0.594 + 1.02i)23-s + (−0.0404 − 0.0701i)25-s + (−0.358 − 0.620i)29-s + (−0.0600 + 0.104i)31-s + 1.27·35-s + 1.07·37-s + (0.274 − 0.475i)41-s + (0.797 + 1.38i)43-s + (0.112 + 0.194i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.926 - 0.377i$
Analytic conductor: \(134.950\)
Root analytic conductor: \(11.6168\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :7/2),\ 0.926 - 0.377i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.521146133\)
\(L(\frac12)\) \(\approx\) \(2.521146133\)
\(L(\frac{9}{2})\) 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 \( 1 + (-145. + 251. i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-555. - 962. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (2.24e3 + 3.88e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (1.21e3 - 2.11e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 1.59e4T + 4.10e8T^{2} \)
19 \( 1 - 4.99e4T + 8.93e8T^{2} \)
23 \( 1 + (3.46e4 - 6.00e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (4.70e4 + 8.14e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (9.96e3 - 1.72e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 3.31e5T + 9.49e10T^{2} \)
41 \( 1 + (-1.21e5 + 2.09e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-4.15e5 - 7.20e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (-8.00e4 - 1.38e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + 3.11e5T + 1.17e12T^{2} \)
59 \( 1 + (-1.56e5 + 2.70e5i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-2.87e4 - 4.97e4i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (2.05e6 - 3.55e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 4.03e5T + 9.09e12T^{2} \)
73 \( 1 + 8.23e5T + 1.10e13T^{2} \)
79 \( 1 + (-4.89e5 - 8.47e5i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-1.85e6 - 3.20e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 2.09e6T + 4.42e13T^{2} \)
97 \( 1 + (1.75e6 + 3.03e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−9.733954829610855283969796279601, −9.164051339568694946989533065222, −8.346930962396929025427184698857, −7.48276981673930096800757979436, −5.86328874523261240290124036110, −5.49543294944908728732056533453, −4.47480725355393445317058611916, −2.97459124780577070285062924930, −1.90083109545392407068944477635, −0.874892504912275850595263827778, 0.60776532691640326125534544486, 1.91085515533966376208433928882, 2.87989651448625127224581480380, 4.17790425299334362443262806943, 5.10123171740779089315764779166, 6.32449022307670826372427869232, 7.29147334774313563740385624975, 7.76906359091204273748761835827, 9.186387595694869102726332458173, 10.20697290965628735709402085061

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