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Properties

Label 2-432-108.83-c1-0-10
Degree $2$
Conductor $432$
Sign $0.270 + 0.962i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.19 + 1.25i)3-s + (0.311 + 0.371i)5-s + (−0.958 − 2.63i)7-s + (−0.140 − 2.99i)9-s + (−4.24 − 3.56i)11-s + (−0.0238 − 0.135i)13-s + (−0.837 − 0.0535i)15-s + (3.57 − 2.06i)17-s + (4.52 + 2.61i)19-s + (4.44 + 1.94i)21-s + (−2.38 − 0.866i)23-s + (0.827 − 4.69i)25-s + (3.92 + 3.40i)27-s + (−2.79 − 0.493i)29-s + (2.14 − 5.89i)31-s + ⋯
L(s)  = 1  + (−0.690 + 0.723i)3-s + (0.139 + 0.166i)5-s + (−0.362 − 0.995i)7-s + (−0.0468 − 0.998i)9-s + (−1.27 − 1.07i)11-s + (−0.00662 − 0.0375i)13-s + (−0.216 − 0.0138i)15-s + (0.868 − 0.501i)17-s + (1.03 + 0.599i)19-s + (0.969 + 0.424i)21-s + (−0.496 − 0.180i)23-s + (0.165 − 0.938i)25-s + (0.754 + 0.655i)27-s + (−0.519 − 0.0916i)29-s + (0.385 − 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612535 - 0.464177i\)
\(L(\frac12)\) \(\approx\) \(0.612535 - 0.464177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 1.25i)T \)
good5 \( 1 + (-0.311 - 0.371i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.958 + 2.63i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (4.24 + 3.56i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.0238 + 0.135i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-3.57 + 2.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.52 - 2.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.38 + 0.866i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (2.79 + 0.493i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.14 + 5.89i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (4.89 + 8.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.97 - 0.877i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.705 + 0.840i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.84 - 0.671i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 + (3.92 - 3.29i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-5.00 + 1.82i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-11.6 + 2.05i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (7.77 + 13.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.66 - 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.12 - 0.374i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (1.26 - 7.15i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.58 - 1.49i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.42 + 3.71i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.73976481079350150142694135543, −10.24647274197282169491953765268, −9.508601051118116159400865990763, −8.149454226107989542980193865515, −7.24285900945925878081438086369, −5.98442821899092836179184424100, −5.33368131376240330235755945915, −4.01618030468638084035696713265, −3.04860133877712328254880639791, −0.52991175079832334638775915138, 1.72955826777404922973192970024, 3.01589876122434991443766247994, 5.09680553868326237567800204840, 5.43508701157619646284496839721, 6.69395509320326203028972671419, 7.55983563784999920536410081686, 8.464352920563201965061167701311, 9.710570540526313310516677642814, 10.39374862633025309223499017363, 11.59782135295340109864192245607

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