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Properties

Label 2-432-16.5-c1-0-10
Degree $2$
Conductor $432$
Sign $0.0381 - 0.999i$
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  + (1.24 + 0.678i)2-s + (1.07 + 1.68i)4-s + (−0.920 − 0.920i)5-s + 4.02i·7-s + (0.196 + 2.82i)8-s + (−0.517 − 1.76i)10-s + (0.00588 + 0.00588i)11-s + (−1.05 + 1.05i)13-s + (−2.72 + 4.98i)14-s + (−1.67 + 3.63i)16-s + 8.16·17-s + (−2.21 + 2.21i)19-s + (0.556 − 2.54i)20-s + (0.00330 + 0.0112i)22-s − 4.29i·23-s + ⋯
L(s)  = 1  + (0.877 + 0.479i)2-s + (0.539 + 0.841i)4-s + (−0.411 − 0.411i)5-s + 1.51i·7-s + (0.0695 + 0.997i)8-s + (−0.163 − 0.558i)10-s + (0.00177 + 0.00177i)11-s + (−0.291 + 0.291i)13-s + (−0.729 + 1.33i)14-s + (−0.417 + 0.908i)16-s + 1.97·17-s + (−0.509 + 0.509i)19-s + (0.124 − 0.568i)20-s + (0.000705 + 0.00240i)22-s − 0.895i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0381 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0381 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0381 - 0.999i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.0381 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52794 + 1.47075i\)
\(L(\frac12)\) \(\approx\) \(1.52794 + 1.47075i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.678i)T \)
3 \( 1 \)
good5 \( 1 + (0.920 + 0.920i)T + 5iT^{2} \)
7 \( 1 - 4.02iT - 7T^{2} \)
11 \( 1 + (-0.00588 - 0.00588i)T + 11iT^{2} \)
13 \( 1 + (1.05 - 1.05i)T - 13iT^{2} \)
17 \( 1 - 8.16T + 17T^{2} \)
19 \( 1 + (2.21 - 2.21i)T - 19iT^{2} \)
23 \( 1 + 4.29iT - 23T^{2} \)
29 \( 1 + (1.74 - 1.74i)T - 29iT^{2} \)
31 \( 1 - 4.60T + 31T^{2} \)
37 \( 1 + (3.07 + 3.07i)T + 37iT^{2} \)
41 \( 1 + 9.43iT - 41T^{2} \)
43 \( 1 + (-6.55 - 6.55i)T + 43iT^{2} \)
47 \( 1 - 2.90T + 47T^{2} \)
53 \( 1 + (8.65 + 8.65i)T + 53iT^{2} \)
59 \( 1 + (6.98 + 6.98i)T + 59iT^{2} \)
61 \( 1 + (0.243 - 0.243i)T - 61iT^{2} \)
67 \( 1 + (-4.55 + 4.55i)T - 67iT^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 + 7.35iT - 73T^{2} \)
79 \( 1 - 8.69T + 79T^{2} \)
83 \( 1 + (-8.93 + 8.93i)T - 83iT^{2} \)
89 \( 1 - 9.17iT - 89T^{2} \)
97 \( 1 - 3.27T + 97T^{2} \)
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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

−11.94720421800633914226559164156, −10.66509973689270380007701695366, −9.356888388123584234579921243195, −8.379151207609078613063931451498, −7.77796746424565019831626505216, −6.42260885059104412083933514330, −5.60835549440267499935759843721, −4.76403148519037754229182681039, −3.49183532012560425921108585561, −2.26833808727344750928150620302, 1.13943247415978004736359037036, 3.07308885334030755256282683766, 3.85144298260746023688143455644, 4.91577858123587753295682767949, 6.09422877214553543262785678930, 7.27130901415841174805327307145, 7.72715979302660675126743306063, 9.572376145299620580019378865434, 10.31082109534328098703102748338, 10.96024963081187417084621532826

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