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Properties

Label 2-432-108.23-c1-0-9
Degree $2$
Conductor $432$
Sign $0.923 - 0.384i$
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  + (1.72 + 0.107i)3-s + (0.902 + 0.159i)5-s + (2.30 + 2.75i)7-s + (2.97 + 0.373i)9-s + (−0.316 − 1.79i)11-s + (−4.18 + 1.52i)13-s + (1.54 + 0.372i)15-s + (−3.92 − 2.26i)17-s + (−0.794 + 0.458i)19-s + (3.69 + 5.00i)21-s + (5.68 + 4.76i)23-s + (−3.90 − 1.42i)25-s + (5.10 + 0.966i)27-s + (3.02 − 8.31i)29-s + (3.84 − 4.58i)31-s + ⋯
L(s)  = 1  + (0.998 + 0.0623i)3-s + (0.403 + 0.0711i)5-s + (0.872 + 1.04i)7-s + (0.992 + 0.124i)9-s + (−0.0952 − 0.540i)11-s + (−1.16 + 0.422i)13-s + (0.398 + 0.0961i)15-s + (−0.951 − 0.549i)17-s + (−0.182 + 0.105i)19-s + (0.806 + 1.09i)21-s + (1.18 + 0.994i)23-s + (−0.781 − 0.284i)25-s + (0.982 + 0.186i)27-s + (0.562 − 1.54i)29-s + (0.690 − 0.823i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.923 - 0.384i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.923 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07613 + 0.415044i\)
\(L(\frac12)\) \(\approx\) \(2.07613 + 0.415044i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.107i)T \)
good5 \( 1 + (-0.902 - 0.159i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2.30 - 2.75i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.316 + 1.79i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (4.18 - 1.52i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (3.92 + 2.26i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.794 - 0.458i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.68 - 4.76i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-3.02 + 8.31i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.84 + 4.58i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (3.15 - 5.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.16 + 3.19i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.919 - 0.162i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.999 - 0.838i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 2.53iT - 53T^{2} \)
59 \( 1 + (-2.13 + 12.1i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (2.66 - 2.23i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.986 + 2.71i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.44 - 7.69i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.528 + 0.915i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.99 - 10.9i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (15.7 + 5.75i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-13.1 + 7.58i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.86 - 10.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.37468670050035133164169328454, −10.01288226557311270917554064320, −9.346752334046700190865255870016, −8.522959679264793516129857320920, −7.76033316433645092010834090487, −6.63203639573946763631543417441, −5.32231274525794745229372477369, −4.39743362992243427115041004576, −2.77641172527978217539394671047, −1.99665153875436383829171625446, 1.55758216004409471268931405600, 2.79879141509256196808521502464, 4.29443833796082017835583114443, 4.98059552118680599236075265833, 6.81526594949820261623512039853, 7.39484323528195029060289104344, 8.387473143625103276147961256669, 9.187989215492897759047628022820, 10.32687055275842228750846594474, 10.69333973196685941900192278776

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