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Properties

Label 2-432-36.31-c2-0-11
Degree $2$
Conductor $432$
Sign $-0.742 + 0.669i$
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  + (3.01 − 5.22i)5-s + (−10.2 + 5.90i)7-s + (5.28 − 3.05i)11-s + (7.44 − 12.9i)13-s − 26.6·17-s − 9.45i·19-s + (−17.2 − 9.96i)23-s + (−5.70 − 9.88i)25-s + (−22.3 − 38.6i)29-s + (−5.42 − 3.13i)31-s + 71.3i·35-s − 6.65·37-s + (−8.82 + 15.2i)41-s + (−20.2 + 11.7i)43-s + (−36.4 + 21.0i)47-s + ⋯
L(s)  = 1  + (0.603 − 1.04i)5-s + (−1.46 + 0.844i)7-s + (0.480 − 0.277i)11-s + (0.572 − 0.992i)13-s − 1.57·17-s − 0.497i·19-s + (−0.750 − 0.433i)23-s + (−0.228 − 0.395i)25-s + (−0.769 − 1.33i)29-s + (−0.174 − 0.101i)31-s + 2.03i·35-s − 0.179·37-s + (−0.215 + 0.372i)41-s + (−0.471 + 0.272i)43-s + (−0.775 + 0.447i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.742 + 0.669i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ -0.742 + 0.669i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.312370 - 0.813134i\)
\(L(\frac12)\) \(\approx\) \(0.312370 - 0.813134i\)
\(L(2)\) 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 \( 1 + (-3.01 + 5.22i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (10.2 - 5.90i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5.28 + 3.05i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.44 + 12.9i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 26.6T + 289T^{2} \)
19 \( 1 + 9.45iT - 361T^{2} \)
23 \( 1 + (17.2 + 9.96i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (22.3 + 38.6i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (5.42 + 3.13i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 6.65T + 1.36e3T^{2} \)
41 \( 1 + (8.82 - 15.2i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (20.2 - 11.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (36.4 - 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 51.6T + 2.80e3T^{2} \)
59 \( 1 + (32.9 + 18.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (45.3 + 78.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (53.4 + 30.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 39.5iT - 5.04e3T^{2} \)
73 \( 1 - 35.0T + 5.32e3T^{2} \)
79 \( 1 + (-77.9 + 45.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-102. + 59.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 14.4T + 7.92e3T^{2} \)
97 \( 1 + (-67.5 - 117. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−10.46920522135091739729157418437, −9.345322979920487566062805144582, −9.099436562408867825278490696063, −8.081279876293334701454744529367, −6.41989754388246318609255547483, −6.02687294645602251487102259650, −4.85631306589111353878175774980, −3.48749025331689467243171543568, −2.17069177567424803298836468717, −0.34396614024225142693688126572, 1.91013990302251043340821721895, 3.33071327245143012087290052744, 4.16628936363199868046473561373, 5.95901501239281055869689911581, 6.76538332977123576678272255863, 7.08334318953932838577355597408, 8.823383744548842675037343930794, 9.597604081058174073252272416591, 10.38162623268147499279299114010, 11.04410575675542486155844486285

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