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Properties

Label 2-432-16.13-c1-0-15
Degree $2$
Conductor $432$
Sign $0.802 + 0.596i$
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  + (−0.936 − 1.05i)2-s + (−0.246 + 1.98i)4-s + (1.12 − 1.12i)5-s + 1.33i·7-s + (2.33 − 1.59i)8-s + (−2.25 − 0.139i)10-s + (−0.386 + 0.386i)11-s + (2.49 + 2.49i)13-s + (1.40 − 1.24i)14-s + (−3.87 − 0.977i)16-s + 5.71·17-s + (−0.755 − 0.755i)19-s + (1.96 + 2.52i)20-s + (0.771 + 0.0477i)22-s − 2.22i·23-s + ⋯
L(s)  = 1  + (−0.662 − 0.749i)2-s + (−0.123 + 0.992i)4-s + (0.505 − 0.505i)5-s + 0.502i·7-s + (0.825 − 0.564i)8-s + (−0.713 − 0.0440i)10-s + (−0.116 + 0.116i)11-s + (0.691 + 0.691i)13-s + (0.376 − 0.332i)14-s + (−0.969 − 0.244i)16-s + 1.38·17-s + (−0.173 − 0.173i)19-s + (0.439 + 0.563i)20-s + (0.164 + 0.0101i)22-s − 0.464i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.802 + 0.596i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.802 + 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07192 - 0.355028i\)
\(L(\frac12)\) \(\approx\) \(1.07192 - 0.355028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.936 + 1.05i)T \)
3 \( 1 \)
good5 \( 1 + (-1.12 + 1.12i)T - 5iT^{2} \)
7 \( 1 - 1.33iT - 7T^{2} \)
11 \( 1 + (0.386 - 0.386i)T - 11iT^{2} \)
13 \( 1 + (-2.49 - 2.49i)T + 13iT^{2} \)
17 \( 1 - 5.71T + 17T^{2} \)
19 \( 1 + (0.755 + 0.755i)T + 19iT^{2} \)
23 \( 1 + 2.22iT - 23T^{2} \)
29 \( 1 + (-0.659 - 0.659i)T + 29iT^{2} \)
31 \( 1 - 9.38T + 31T^{2} \)
37 \( 1 + (-5.75 + 5.75i)T - 37iT^{2} \)
41 \( 1 + 9.62iT - 41T^{2} \)
43 \( 1 + (0.132 - 0.132i)T - 43iT^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (-2.31 + 2.31i)T - 53iT^{2} \)
59 \( 1 + (7.72 - 7.72i)T - 59iT^{2} \)
61 \( 1 + (6.74 + 6.74i)T + 61iT^{2} \)
67 \( 1 + (3.59 + 3.59i)T + 67iT^{2} \)
71 \( 1 - 9.17iT - 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + (1.58 + 1.58i)T + 83iT^{2} \)
89 \( 1 - 9.68iT - 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−11.00382698745192063149036663335, −10.07038011629259378352928747303, −9.300019754964941129758588070169, −8.602366535430323273000296188998, −7.68472533911916181433763546794, −6.41365832126966231292143689601, −5.20608710869166060357692674283, −3.93967471535559919910756144606, −2.59207587385633929661411297980, −1.27418653111544083232606323555, 1.19512141464688645166808363116, 3.04569102044393463194892868157, 4.68632345265092911412484273435, 5.93279118237632170510153881209, 6.47468147626584311640742263318, 7.76414140142311638518905826851, 8.221720625663628258675982559808, 9.598703491587866553138440382855, 10.17305901760948947947830309224, 10.84391946548119961940577662927

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