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Properties

Label 2-432-27.4-c1-0-13
Degree $2$
Conductor $432$
Sign $0.792 + 0.609i$
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  + (1.72 − 0.192i)3-s + (0.0709 − 0.402i)5-s + (2.76 − 2.31i)7-s + (2.92 − 0.661i)9-s + (−0.933 − 5.29i)11-s + (−6.08 + 2.21i)13-s + (0.0448 − 0.706i)15-s + (−1.42 + 2.47i)17-s + (2.71 + 4.70i)19-s + (4.30 − 4.51i)21-s + (−2.18 − 1.83i)23-s + (4.54 + 1.65i)25-s + (4.90 − 1.70i)27-s + (2.41 + 0.877i)29-s + (2.53 + 2.12i)31-s + ⋯
L(s)  = 1  + (0.993 − 0.110i)3-s + (0.0317 − 0.179i)5-s + (1.04 − 0.875i)7-s + (0.975 − 0.220i)9-s + (−0.281 − 1.59i)11-s + (−1.68 + 0.614i)13-s + (0.0115 − 0.182i)15-s + (−0.346 + 0.600i)17-s + (0.623 + 1.07i)19-s + (0.939 − 0.985i)21-s + (−0.455 − 0.381i)23-s + (0.908 + 0.330i)25-s + (0.944 − 0.327i)27-s + (0.447 + 0.162i)29-s + (0.455 + 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 + 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.792 + 0.609i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.792 + 0.609i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91024 - 0.649605i\)
\(L(\frac12)\) \(\approx\) \(1.91024 - 0.649605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.192i)T \)
good5 \( 1 + (-0.0709 + 0.402i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (-2.76 + 2.31i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (0.933 + 5.29i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (6.08 - 2.21i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (1.42 - 2.47i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.71 - 4.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.18 + 1.83i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.41 - 0.877i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-2.53 - 2.12i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (0.462 - 0.801i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.40 + 1.23i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.154 - 0.873i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (7.89 - 6.62i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 4.56T + 53T^{2} \)
59 \( 1 + (1.06 - 6.02i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (4.86 - 4.07i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-5.49 + 2.00i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-5.61 + 9.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.59 - 6.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.0666 - 0.0242i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (15.4 + 5.63i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-3.42 - 5.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.10 - 11.9i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.92046261319057470828743848596, −10.16018132015359911838554926023, −9.128281218340934554126076876875, −8.154667783750347882425354453297, −7.70774436198883010858051024935, −6.56643879172519329632417809678, −5.05153403416406958659462769957, −4.11025683217604836846266783261, −2.86009511396582083454835321141, −1.38880041455567092114465172154, 2.11558091796085023552989316474, 2.76719239716343027460509545862, 4.70741814823472741205575792790, 5.01052589297705372835376296394, 6.98160698065538808521097880240, 7.60641700562894871263544554865, 8.471623355721155735214239138881, 9.585911849976596903615545293946, 9.966682414720513761470666637526, 11.29724420746548031059480951703

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