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Properties

Label 2-432-27.13-c1-0-3
Degree $2$
Conductor $432$
Sign $0.300 - 0.953i$
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  + (−1.59 + 0.678i)3-s + (1.47 + 1.23i)5-s + (0.495 + 0.180i)7-s + (2.07 − 2.16i)9-s + (1.04 − 0.877i)11-s + (−0.231 + 1.31i)13-s + (−3.18 − 0.970i)15-s + (−1.45 + 2.51i)17-s + (4.12 + 7.14i)19-s + (−0.912 + 0.0489i)21-s + (−4.64 + 1.68i)23-s + (−0.226 − 1.28i)25-s + (−1.84 + 4.85i)27-s + (1.56 + 8.84i)29-s + (7.35 − 2.67i)31-s + ⋯
L(s)  = 1  + (−0.920 + 0.391i)3-s + (0.658 + 0.552i)5-s + (0.187 + 0.0682i)7-s + (0.692 − 0.720i)9-s + (0.315 − 0.264i)11-s + (−0.0642 + 0.364i)13-s + (−0.822 − 0.250i)15-s + (−0.352 + 0.609i)17-s + (0.946 + 1.63i)19-s + (−0.199 + 0.0106i)21-s + (−0.968 + 0.352i)23-s + (−0.0452 − 0.256i)25-s + (−0.355 + 0.934i)27-s + (0.289 + 1.64i)29-s + (1.32 − 0.481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.300 - 0.953i$
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: \((2,\ 432,\ (\ :1/2),\ 0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.925991 + 0.679424i\)
\(L(\frac12)\) \(\approx\) \(0.925991 + 0.679424i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.678i)T \)
good5 \( 1 + (-1.47 - 1.23i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.495 - 0.180i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.04 + 0.877i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.231 - 1.31i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.45 - 2.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.12 - 7.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.64 - 1.68i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.56 - 8.84i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-7.35 + 2.67i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.567 + 0.982i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.675 - 3.82i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.69 + 1.42i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-6.47 - 2.35i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 8.26T + 53T^{2} \)
59 \( 1 + (1.66 + 1.39i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-7.98 - 2.90i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.319 + 1.80i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (3.51 - 6.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.42 + 4.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.99 + 11.3i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (2.49 + 14.1i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.50 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.2 + 11.9i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.36340273245074175161332392976, −10.23432512728956554852552467580, −9.977507813139570732416616479006, −8.747667162603466155473243843570, −7.50008169781253788979913788707, −6.29289471984360304175131268179, −5.87785838206944413134307698862, −4.61285114815031641951709991702, −3.44053038613313334265484437606, −1.62868031342975493214555577270, 0.898208178567494305170188534972, 2.42219190444328911191588217002, 4.46434002688376331077291303091, 5.21611870832045037219985620862, 6.20712106482283736820638409621, 7.10750043321769821769746875282, 8.131487571678810122990944843621, 9.377636943649822228854142310442, 10.02313603085809416653316528153, 11.15903500539557840719007515178

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