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Properties

Label 2-432-108.95-c1-0-8
Degree $2$
Conductor $432$
Sign $0.479 - 0.877i$
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  + (0.573 + 1.63i)3-s + (2.37 − 2.82i)5-s + (−1.65 + 4.53i)7-s + (−2.34 + 1.87i)9-s + (1.42 − 1.19i)11-s + (−0.547 + 3.10i)13-s + (5.98 + 2.25i)15-s + (4.94 + 2.85i)17-s + (4.12 − 2.37i)19-s + (−8.36 − 0.0977i)21-s + (−2.05 + 0.749i)23-s + (−1.50 − 8.51i)25-s + (−4.40 − 2.75i)27-s + (−1.39 + 0.245i)29-s + (−0.634 − 1.74i)31-s + ⋯
L(s)  = 1  + (0.331 + 0.943i)3-s + (1.06 − 1.26i)5-s + (−0.624 + 1.71i)7-s + (−0.780 + 0.624i)9-s + (0.429 − 0.360i)11-s + (−0.151 + 0.861i)13-s + (1.54 + 0.583i)15-s + (1.19 + 0.692i)17-s + (0.945 − 0.545i)19-s + (−1.82 − 0.0213i)21-s + (−0.429 + 0.156i)23-s + (−0.300 − 1.70i)25-s + (−0.847 − 0.530i)27-s + (−0.258 + 0.0456i)29-s + (−0.113 − 0.313i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.479 - 0.877i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.479 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47634 + 0.875263i\)
\(L(\frac12)\) \(\approx\) \(1.47634 + 0.875263i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.573 - 1.63i)T \)
good5 \( 1 + (-2.37 + 2.82i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (1.65 - 4.53i)T + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (-1.42 + 1.19i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.547 - 3.10i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.94 - 2.85i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.12 + 2.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.05 - 0.749i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (1.39 - 0.245i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (0.634 + 1.74i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.01 + 3.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.96 - 0.523i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-4.32 - 5.15i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (6.36 + 2.31i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 - 3.09iT - 53T^{2} \)
59 \( 1 + (6.96 + 5.84i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.99 + 1.08i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-3.41 - 0.602i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-6.15 + 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.95 + 6.84i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (12.7 - 2.24i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.606 + 3.44i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-8.76 + 5.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.536 + 0.450i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.40754163084038012619213382875, −9.936662685370260432633018822456, −9.298479461111945461403083545019, −9.074679345141553098936546317193, −8.067782568029112460692566179960, −6.04824492583142852841062493024, −5.63552486871995651475378680620, −4.66829174711888425118714642335, −3.20611506442904194028723511819, −1.91597266890155306512659367374, 1.19801391093826202556837554527, 2.81738477347037152699159129459, 3.61220091890962336665725410935, 5.61548169567096876831801162921, 6.54040213379007752561150792445, 7.26468034373266629397487642847, 7.78493283085205538429238886483, 9.571189081975737218832391062691, 10.01482475679488918568051076427, 10.81118837770499695491522962143

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