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Properties

Label 2-432-144.77-c2-0-40
Degree $2$
Conductor $432$
Sign $-0.370 + 0.928i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.03 − 1.71i)2-s + (−1.85 − 3.54i)4-s + (7.78 − 2.08i)5-s + (2.60 − 1.50i)7-s + (−7.98 − 0.483i)8-s + (4.48 − 15.4i)10-s + (1.54 − 5.77i)11-s + (14.7 − 3.94i)13-s + (0.121 − 6.02i)14-s + (−9.08 + 13.1i)16-s + 23.3i·17-s + (−8.30 − 8.30i)19-s + (−21.8 − 23.7i)20-s + (−8.29 − 8.63i)22-s + (2.08 − 3.60i)23-s + ⋯
L(s)  = 1  + (0.517 − 0.855i)2-s + (−0.464 − 0.885i)4-s + (1.55 − 0.417i)5-s + (0.372 − 0.215i)7-s + (−0.998 − 0.0604i)8-s + (0.448 − 1.54i)10-s + (0.140 − 0.525i)11-s + (1.13 − 0.303i)13-s + (0.00866 − 0.430i)14-s + (−0.568 + 0.822i)16-s + 1.37i·17-s + (−0.437 − 0.437i)19-s + (−1.09 − 1.18i)20-s + (−0.376 − 0.392i)22-s + (0.0905 − 0.156i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.370 + 0.928i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ -0.370 + 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60943 - 2.37601i\)
\(L(\frac12)\) \(\approx\) \(1.60943 - 2.37601i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 + 1.71i)T \)
3 \( 1 \)
good5 \( 1 + (-7.78 + 2.08i)T + (21.6 - 12.5i)T^{2} \)
7 \( 1 + (-2.60 + 1.50i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-1.54 + 5.77i)T + (-104. - 60.5i)T^{2} \)
13 \( 1 + (-14.7 + 3.94i)T + (146. - 84.5i)T^{2} \)
17 \( 1 - 23.3iT - 289T^{2} \)
19 \( 1 + (8.30 + 8.30i)T + 361iT^{2} \)
23 \( 1 + (-2.08 + 3.60i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (24.6 + 6.59i)T + (728. + 420.5i)T^{2} \)
31 \( 1 + (-11.4 + 19.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-27.8 + 27.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (36.7 - 63.6i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (70.9 + 19.0i)T + (1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-39.5 + 22.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-21.0 - 21.0i)T + 2.80e3iT^{2} \)
59 \( 1 + (77.1 - 20.6i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-13.6 + 51.0i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-44.8 + 12.0i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 82.1T + 5.04e3T^{2} \)
73 \( 1 - 25.3iT - 5.32e3T^{2} \)
79 \( 1 + (-45.7 - 79.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-45.3 - 12.1i)T + (5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 79.4T + 7.92e3T^{2} \)
97 \( 1 + (57.7 + 100. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.70949577213197850019705158624, −9.965515694750794077999133799181, −9.043403901501407300917483002342, −8.302258978232215616706111342616, −6.27663561388326795302905497935, −5.90278403252176484817133470092, −4.78360322862518802236804312541, −3.58724645159552434243733063827, −2.12328323970038030503630306974, −1.16222336721602159729806950177, 1.88616933801804316990178565579, 3.24871334381774451772427228782, 4.74216724019598985878130860871, 5.62438840833444300571324283038, 6.43885640546715749278637092869, 7.20617325387224064263526016669, 8.517956337436999436645052861622, 9.280272671512050148872935541222, 10.13542802109790428907236223336, 11.31435245874015750384844048063

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