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Properties

Label 2-432-108.59-c1-0-0
Degree $2$
Conductor $432$
Sign $-0.823 - 0.566i$
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  + (−0.796 − 1.53i)3-s + (1.12 + 3.08i)5-s + (−3.05 + 0.537i)7-s + (−1.73 + 2.45i)9-s + (−1.75 − 0.637i)11-s + (−3.44 − 2.88i)13-s + (3.84 − 4.18i)15-s + (−5.33 − 3.07i)17-s + (−5.61 + 3.24i)19-s + (3.25 + 4.26i)21-s + (−0.0995 + 0.564i)23-s + (−4.42 + 3.71i)25-s + (5.14 + 0.711i)27-s + (5.05 + 6.02i)29-s + (6.55 + 1.15i)31-s + ⋯
L(s)  = 1  + (−0.459 − 0.888i)3-s + (0.502 + 1.37i)5-s + (−1.15 + 0.203i)7-s + (−0.577 + 0.816i)9-s + (−0.527 − 0.192i)11-s + (−0.954 − 0.800i)13-s + (0.994 − 1.08i)15-s + (−1.29 − 0.746i)17-s + (−1.28 + 0.744i)19-s + (0.710 + 0.930i)21-s + (−0.0207 + 0.117i)23-s + (−0.884 + 0.742i)25-s + (0.990 + 0.136i)27-s + (0.938 + 1.11i)29-s + (1.17 + 0.207i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.823 - 0.566i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.823 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0765620 + 0.246288i\)
\(L(\frac12)\) \(\approx\) \(0.0765620 + 0.246288i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.796 + 1.53i)T \)
good5 \( 1 + (-1.12 - 3.08i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (3.05 - 0.537i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (1.75 + 0.637i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (3.44 + 2.88i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (5.33 + 3.07i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.61 - 3.24i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0995 - 0.564i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-5.05 - 6.02i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-6.55 - 1.15i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (2.51 - 4.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.01 + 4.78i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (3.06 - 8.40i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (1.29 + 7.32i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 4.40iT - 53T^{2} \)
59 \( 1 + (1.19 - 0.436i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.757 - 4.29i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (3.35 - 3.99i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (-2.77 + 4.80i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.12 + 8.87i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.530 + 0.631i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.90 - 4.95i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (2.31 - 1.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.55 - 3.47i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.50582370938739225248057377749, −10.41838446091794388408871558965, −10.17785537567579284395478702528, −8.721280991183647729090833803412, −7.55761560805007254924772012314, −6.57910463708238791835909243319, −6.33805694297981971639565733427, −5.04260892872809820259636443580, −3.01672267209113261627821257547, −2.39659457430313423034196819540, 0.15585611303037120059380402719, 2.45366767719177596114005614002, 4.30888529417268030841026478208, 4.68106752973436450426164999581, 5.99991452051450808187558260734, 6.71503755382601999955798569706, 8.445635523891267381108347208163, 9.154691264667166566349830602078, 9.826714695913490234230723243224, 10.56176778799139826749179667053

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