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Properties

Label 2-432-144.85-c1-0-16
Degree $2$
Conductor $432$
Sign $0.675 - 0.737i$
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 + i)2-s + 2i·4-s + (1.86 − 0.5i)5-s + (3.86 − 2.23i)7-s + (−2 + 2i)8-s + (2.36 + 1.36i)10-s + (0.5 − 1.86i)11-s + (−0.598 − 2.23i)13-s + (6.09 + 1.63i)14-s − 4·16-s − 4·17-s + (−3 + 3i)19-s + (1 + 3.73i)20-s + (2.36 − 1.36i)22-s + (5.59 + 3.23i)23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (0.834 − 0.223i)5-s + (1.46 − 0.843i)7-s + (−0.707 + 0.707i)8-s + (0.748 + 0.431i)10-s + (0.150 − 0.562i)11-s + (−0.165 − 0.619i)13-s + (1.62 + 0.436i)14-s − 16-s − 0.970·17-s + (−0.688 + 0.688i)19-s + (0.223 + 0.834i)20-s + (0.504 − 0.291i)22-s + (1.16 + 0.673i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.675 - 0.737i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21691 + 0.975463i\)
\(L(\frac12)\) \(\approx\) \(2.21691 + 0.975463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
good5 \( 1 + (-1.86 + 0.5i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.86 + 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 1.86i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.598 + 2.23i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + (3 - 3i)T - 19iT^{2} \)
23 \( 1 + (-5.59 - 3.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.866 - 0.232i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.59 - 7.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.26 + 4.26i)T + 37iT^{2} \)
41 \( 1 + (0.696 + 0.401i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.69 - 6.33i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.598 - 1.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.73 + 5.73i)T + 53iT^{2} \)
59 \( 1 + (-1.5 + 0.401i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.13 + 0.571i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.23 + 8.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + 7.46iT - 73T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.1 - 3.79i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 15.8iT - 89T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.17881796868958446749845448641, −10.68374648067048611888876791469, −9.192303429960026292072068049187, −8.346760097979251694027590529866, −7.52225300487081099952851990246, −6.51473834886394165267783479073, −5.38765840890199586588349505677, −4.75306077083301007876810933407, −3.50438069768030824563392271604, −1.77942955101796915401759718895, 1.85623255560960312270627319693, 2.44731318008402353571952307823, 4.36342768056604405552351597458, 5.02471312083111469700872889468, 6.07125515995286220928475998199, 7.04985606924853834159894697493, 8.668149180052924758769512672529, 9.275541665375973651252760629052, 10.38934402508992412156793789112, 11.21094042012757561568322976825

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