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Properties

Label 2-432-144.13-c1-0-19
Degree $2$
Conductor $432$
Sign $-0.461 + 0.887i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (−1.73 + i)4-s + (−0.5 − 1.86i)5-s + (−3.86 + 2.23i)7-s + (−2 − 1.99i)8-s + (2.36 − 1.36i)10-s + (−1.86 − 0.5i)11-s + (2.23 − 0.598i)13-s + (−4.46 − 4.46i)14-s + (1.99 − 3.46i)16-s − 4·17-s + (−3 − 3i)19-s + (2.73 + 2.73i)20-s − 2.73i·22-s + (−5.59 − 3.23i)23-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.5i)4-s + (−0.223 − 0.834i)5-s + (−1.46 + 0.843i)7-s + (−0.707 − 0.707i)8-s + (0.748 − 0.431i)10-s + (−0.562 − 0.150i)11-s + (0.619 − 0.165i)13-s + (−1.19 − 1.19i)14-s + (0.499 − 0.866i)16-s − 0.970·17-s + (−0.688 − 0.688i)19-s + (0.610 + 0.610i)20-s − 0.582i·22-s + (−1.16 − 0.673i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.461 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.461 + 0.887i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.461 + 0.887i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 \)
good5 \( 1 + (0.5 + 1.86i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (3.86 - 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.86 + 0.5i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.23 + 0.598i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (3 + 3i)T + 19iT^{2} \)
23 \( 1 + (5.59 + 3.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.232 - 0.866i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (4.59 - 7.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 - 4.26i)T - 37iT^{2} \)
41 \( 1 + (-0.696 - 0.401i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.33 - 1.69i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.598 - 1.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.73 - 5.73i)T - 53iT^{2} \)
59 \( 1 + (0.401 + 1.5i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.571 + 2.13i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-8.33 + 2.23i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 - 7.46iT - 73T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.79 - 14.1i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 15.8iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.77946955005253783175917830390, −9.583934451474849852673443506519, −8.769601167982884738584657984208, −8.346419816867172562756666292404, −6.87357772999912523565986606532, −6.18638213305003710235403193918, −5.22732228039679318530847607826, −4.13717858132535087747239390937, −2.86441970691719051468980188621, 0, 2.25448936337515098028431852743, 3.53876643103120930119383344363, 4.07702911576728442412049536028, 5.80690873543157151881214330064, 6.64534231149772874591289278354, 7.76392532062814995024046768440, 9.098307607041205100758960962431, 9.933342847714583351780465831709, 10.64013994313423445834184142874

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