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Properties

Label 2-432-27.13-c1-0-12
Degree $2$
Conductor $432$
Sign $0.391 + 0.920i$
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  + (0.198 + 1.72i)3-s + (−2.79 − 2.34i)5-s + (−0.843 − 0.306i)7-s + (−2.92 + 0.683i)9-s + (4.12 − 3.45i)11-s + (0.560 − 3.17i)13-s + (3.48 − 5.28i)15-s + (3.01 − 5.22i)17-s + (1.24 + 2.15i)19-s + (0.360 − 1.51i)21-s + (−3.10 + 1.13i)23-s + (1.44 + 8.22i)25-s + (−1.75 − 4.89i)27-s + (−1.45 − 8.27i)29-s + (−6.00 + 2.18i)31-s + ⋯
L(s)  = 1  + (0.114 + 0.993i)3-s + (−1.25 − 1.05i)5-s + (−0.318 − 0.116i)7-s + (−0.973 + 0.227i)9-s + (1.24 − 1.04i)11-s + (0.155 − 0.881i)13-s + (0.899 − 1.36i)15-s + (0.731 − 1.26i)17-s + (0.284 + 0.493i)19-s + (0.0787 − 0.329i)21-s + (−0.647 + 0.235i)23-s + (0.289 + 1.64i)25-s + (−0.337 − 0.941i)27-s + (−0.271 − 1.53i)29-s + (−1.07 + 0.392i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771298 - 0.509798i\)
\(L(\frac12)\) \(\approx\) \(0.771298 - 0.509798i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.198 - 1.72i)T \)
good5 \( 1 + (2.79 + 2.34i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.843 + 0.306i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-4.12 + 3.45i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.560 + 3.17i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.01 + 5.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.24 - 2.15i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.10 - 1.13i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.45 + 8.27i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (6.00 - 2.18i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.854 + 1.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.766 - 4.34i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.679 + 0.569i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.22 - 0.809i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 3.93T + 53T^{2} \)
59 \( 1 + (9.65 + 8.10i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-5.55 - 2.02i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.25 + 7.11i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-0.922 + 1.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.38 - 2.39i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.942 - 5.34i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.46 - 8.33i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (6.85 + 11.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.16 - 6.84i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.22677291466746768064171026032, −9.894910668175111247148692368840, −9.196741819111803132681876313166, −8.331408084551520274740057861551, −7.64643920433233097823046092901, −5.98654176832240488487835875751, −5.05006445944065265796241449607, −3.89418153072631053365974666212, −3.36550579286000137257244986457, −0.60397219763270157480931319475, 1.76204522688275956172224177074, 3.30290807474650831235262912092, 4.14948781836145301920252119204, 6.02586573836883329486213635543, 6.97437418517153325296578398173, 7.30821702058469359631453788596, 8.432461621976370393290113673466, 9.380118700114253732662915872285, 10.67233810753990354204069219504, 11.51521874364186402846684249191

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