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Properties

Label 2-432-108.11-c1-0-10
Degree $2$
Conductor $432$
Sign $0.749 - 0.662i$
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  + (1.67 + 0.450i)3-s + (−0.436 + 1.19i)5-s + (3.53 + 0.622i)7-s + (2.59 + 1.50i)9-s + (−4.59 + 1.67i)11-s + (1.75 − 1.47i)13-s + (−1.27 + 1.80i)15-s + (0.393 − 0.227i)17-s + (−5.43 − 3.13i)19-s + (5.62 + 2.63i)21-s + (0.629 + 3.57i)23-s + (2.58 + 2.16i)25-s + (3.66 + 3.68i)27-s + (6.09 − 7.26i)29-s + (−0.352 + 0.0621i)31-s + ⋯
L(s)  = 1  + (0.965 + 0.260i)3-s + (−0.195 + 0.536i)5-s + (1.33 + 0.235i)7-s + (0.864 + 0.502i)9-s + (−1.38 + 0.504i)11-s + (0.485 − 0.407i)13-s + (−0.327 + 0.467i)15-s + (0.0954 − 0.0551i)17-s + (−1.24 − 0.719i)19-s + (1.22 + 0.574i)21-s + (0.131 + 0.744i)23-s + (0.516 + 0.433i)25-s + (0.704 + 0.709i)27-s + (1.13 − 1.34i)29-s + (−0.0633 + 0.0111i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.749 - 0.662i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86163 + 0.704459i\)
\(L(\frac12)\) \(\approx\) \(1.86163 + 0.704459i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 - 0.450i)T \)
good5 \( 1 + (0.436 - 1.19i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-3.53 - 0.622i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (4.59 - 1.67i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.75 + 1.47i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.393 + 0.227i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.43 + 3.13i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.629 - 3.57i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.09 + 7.26i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.352 - 0.0621i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-2.46 - 4.26i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.66 + 5.56i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.37 + 6.52i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.475 - 2.69i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 4.30iT - 53T^{2} \)
59 \( 1 + (9.78 + 3.56i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-2.68 + 15.2i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (1.26 + 1.50i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.92 + 5.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.44 - 5.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.89 + 7.02i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-9.97 - 8.37i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-0.480 - 0.277i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.40 - 1.96i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.88398056697983943286338703373, −10.57837605260982536841522394578, −9.395747755060894883924509011758, −8.199549536851419458080022874201, −7.981264015123543203180145569745, −6.84317001676177985649114043604, −5.23837266575876239500068307708, −4.42980689163213690327614284347, −3.01322170649520032133485017742, −2.00696654131437934645745693873, 1.39786556105037928799990579583, 2.73598262697316758567041504428, 4.20971563285830507464556368492, 5.01412657375530296945751290145, 6.48920249700659495687311891931, 7.74725521199443344117796725581, 8.369023371755169396554913401732, 8.767693157885660872510166772345, 10.30860721136147846363523929611, 10.86001191461487955502219968662

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