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Properties

Label 2-432-9.2-c2-0-9
Degree $2$
Conductor $432$
Sign $-0.800 + 0.598i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.05 − 1.18i)5-s + (−4.05 + 7.02i)7-s + (−17.6 − 10.1i)11-s + (−3.05 − 5.29i)13-s − 17.9i·17-s − 9.11·19-s + (29.0 − 16.7i)23-s + (−9.67 + 16.7i)25-s + (−14.4 − 8.31i)29-s + (−11.1 − 19.3i)31-s + 19.2i·35-s − 50.4·37-s + (−29.9 + 17.3i)41-s + (11.5 − 19.9i)43-s + (−33.1 − 19.1i)47-s + ⋯
L(s)  = 1  + (0.411 − 0.237i)5-s + (−0.579 + 1.00i)7-s + (−1.60 − 0.924i)11-s + (−0.235 − 0.407i)13-s − 1.05i·17-s − 0.479·19-s + (1.26 − 0.729i)23-s + (−0.387 + 0.670i)25-s + (−0.496 − 0.286i)29-s + (−0.360 − 0.624i)31-s + 0.551i·35-s − 1.36·37-s + (−0.730 + 0.421i)41-s + (0.267 − 0.463i)43-s + (−0.705 − 0.407i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.800 + 0.598i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ -0.800 + 0.598i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.164033 - 0.493409i\)
\(L(\frac12)\) \(\approx\) \(0.164033 - 0.493409i\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-2.05 + 1.18i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (4.05 - 7.02i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (17.6 + 10.1i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (3.05 + 5.29i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 17.9iT - 289T^{2} \)
19 \( 1 + 9.11T + 361T^{2} \)
23 \( 1 + (-29.0 + 16.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (14.4 + 8.31i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (11.1 + 19.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 50.4T + 1.36e3T^{2} \)
41 \( 1 + (29.9 - 17.3i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-11.5 + 19.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (33.1 + 19.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 19.0iT - 2.80e3T^{2} \)
59 \( 1 + (2.96 - 1.71i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 + 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (3.14 + 5.45i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 35.9iT - 5.04e3T^{2} \)
73 \( 1 - 47.3T + 5.32e3T^{2} \)
79 \( 1 + (42.2 - 73.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (33.1 + 19.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 + (40.3 - 69.9i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.57943134492028442830487929702, −9.604423012717981049494242312171, −8.828587322506198387686963673091, −7.954462800412299223774631351442, −6.76275454506900782842984875629, −5.54443163501940799332103508261, −5.15614356859366847743402684494, −3.22196477421003218479515465631, −2.36653335536625783726019126306, −0.20139520616062212223795453517, 1.87319335159163778490913242882, 3.24280240460918905607988370041, 4.49869597471401759577878814414, 5.57447918962639702510055295503, 6.83578039756376643305952054491, 7.40751368444858164295824442027, 8.557647380330883618915903576274, 9.792648264645538219393866574827, 10.34696537304586190932083472877, 10.97590133581524301361476328776

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