Properties

Label 2-432-108.11-c1-0-17
Degree $2$
Conductor $432$
Sign $-0.864 + 0.503i$
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  + (0.265 − 1.71i)3-s + (0.601 − 1.65i)5-s + (−3.31 − 0.584i)7-s + (−2.85 − 0.908i)9-s + (−1.91 + 0.696i)11-s + (3.10 − 2.60i)13-s + (−2.66 − 1.46i)15-s + (−2.93 + 1.69i)17-s + (−2.04 − 1.17i)19-s + (−1.88 + 5.52i)21-s + (−0.695 − 3.94i)23-s + (1.46 + 1.22i)25-s + (−2.31 + 4.65i)27-s + (1.89 − 2.25i)29-s + (4.69 − 0.827i)31-s + ⋯
L(s)  = 1  + (0.153 − 0.988i)3-s + (0.268 − 0.738i)5-s + (−1.25 − 0.221i)7-s + (−0.953 − 0.302i)9-s + (−0.577 + 0.210i)11-s + (0.862 − 0.723i)13-s + (−0.688 − 0.378i)15-s + (−0.710 + 0.410i)17-s + (−0.468 − 0.270i)19-s + (−0.410 + 1.20i)21-s + (−0.145 − 0.822i)23-s + (0.292 + 0.245i)25-s + (−0.445 + 0.895i)27-s + (0.351 − 0.418i)29-s + (0.842 − 0.148i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.864 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.864 + 0.503i$
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: \((2,\ 432,\ (\ :1/2),\ -0.864 + 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259614 - 0.961644i\)
\(L(\frac12)\) \(\approx\) \(0.259614 - 0.961644i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.265 + 1.71i)T \)
good5 \( 1 + (-0.601 + 1.65i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (3.31 + 0.584i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (1.91 - 0.696i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-3.10 + 2.60i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.93 - 1.69i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.04 + 1.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.695 + 3.94i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.89 + 2.25i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.69 + 0.827i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.41 - 7.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.67 + 7.95i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (1.29 + 3.55i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.19 + 12.4i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 + (6.19 + 2.25i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (0.0727 - 0.412i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-7.81 - 9.30i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.57 - 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.87 + 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.00 + 5.97i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-12.5 - 10.5i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (2.81 + 1.62i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.7 + 3.90i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.71577275401514080584282907209, −9.869369745883072535793135149099, −8.680343367463443719617803507038, −8.244148815372783730040128971486, −6.83630236267334026212864039170, −6.28867696584037240134700409028, −5.14581301363174854779304585383, −3.57499681144557504339005167883, −2.31588276772709405221883656356, −0.59617609884083701834520281031, 2.61596677244803685813330593183, 3.44061317390885110880531833849, 4.63443801298924821624447732029, 6.04334043498032492788435108615, 6.54906833081690978944706258101, 8.019833573545838461153539154723, 9.148925846925365547340505361989, 9.648202801998353176737819346637, 10.67245431130992563720510582305, 11.12885192795181272072563375253

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