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Properties

Label 2-432-108.95-c1-0-16
Degree $2$
Conductor $432$
Sign $0.911 + 0.411i$
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  + (1.63 + 0.570i)3-s + (2.34 − 2.79i)5-s + (0.550 − 1.51i)7-s + (2.34 + 1.86i)9-s + (−2.21 + 1.85i)11-s + (0.121 − 0.690i)13-s + (5.43 − 3.23i)15-s + (−5.56 − 3.21i)17-s + (−1.62 + 0.938i)19-s + (1.76 − 2.16i)21-s + (−2.18 + 0.794i)23-s + (−1.44 − 8.20i)25-s + (2.78 + 4.38i)27-s + (5.54 − 0.978i)29-s + (3.63 + 9.99i)31-s + ⋯
L(s)  = 1  + (0.944 + 0.329i)3-s + (1.04 − 1.25i)5-s + (0.208 − 0.571i)7-s + (0.783 + 0.621i)9-s + (−0.667 + 0.560i)11-s + (0.0337 − 0.191i)13-s + (1.40 − 0.835i)15-s + (−1.34 − 0.778i)17-s + (−0.373 + 0.215i)19-s + (0.384 − 0.471i)21-s + (−0.455 + 0.165i)23-s + (−0.289 − 1.64i)25-s + (0.535 + 0.844i)27-s + (1.03 − 0.181i)29-s + (0.653 + 1.79i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.911 + 0.411i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.911 + 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09584 - 0.451616i\)
\(L(\frac12)\) \(\approx\) \(2.09584 - 0.451616i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 0.570i)T \)
good5 \( 1 + (-2.34 + 2.79i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.550 + 1.51i)T + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.21 - 1.85i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.121 + 0.690i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (5.56 + 3.21i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.62 - 0.938i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.18 - 0.794i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-5.54 + 0.978i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-3.63 - 9.99i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (4.10 - 7.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.18 - 1.09i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-5.65 - 6.74i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (1.67 + 0.609i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 0.849iT - 53T^{2} \)
59 \( 1 + (4.54 + 3.81i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (9.95 + 3.62i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (14.5 + 2.56i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-5.80 + 10.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.12 - 5.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.47 + 0.789i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.28 + 7.27i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (6.02 - 3.47i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.35 + 1.13i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.72376210410514743927579853781, −10.04889314756482688068290261088, −9.231616284294809438867677581112, −8.545843477701692090826134588288, −7.65234823615563754819554489849, −6.40478301672805050784840974378, −4.86978824816292804559987727872, −4.55445933063841124685847195593, −2.75113862625602835014817006271, −1.53637803981903939366331335788, 2.17911655431833379577082702834, 2.64299708930509391952470317501, 4.11581408124583041263902087452, 5.84990589506302433245291477393, 6.49151758221773552998559762942, 7.53050867225666770321693031216, 8.585234111726979312025998601644, 9.295203622882849517793201567332, 10.38572127214526818938905226044, 10.91213727691729045742494295259

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