Properties

Label 2-432-27.25-c1-0-6
Degree $2$
Conductor $432$
Sign $0.640 - 0.767i$
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  + (1.28 + 1.16i)3-s + (−1.03 + 0.871i)5-s + (4.32 − 1.57i)7-s + (0.296 + 2.98i)9-s + (−0.700 − 0.587i)11-s + (0.939 + 5.32i)13-s + (−2.34 − 0.0886i)15-s + (−3.81 − 6.60i)17-s + (0.825 − 1.42i)19-s + (7.38 + 3.00i)21-s + (5.39 + 1.96i)23-s + (−0.549 + 3.11i)25-s + (−3.09 + 4.17i)27-s + (−0.698 + 3.96i)29-s + (2.21 + 0.806i)31-s + ⋯
L(s)  = 1  + (0.741 + 0.671i)3-s + (−0.464 + 0.389i)5-s + (1.63 − 0.595i)7-s + (0.0987 + 0.995i)9-s + (−0.211 − 0.177i)11-s + (0.260 + 1.47i)13-s + (−0.605 − 0.0229i)15-s + (−0.925 − 1.60i)17-s + (0.189 − 0.328i)19-s + (1.61 + 0.656i)21-s + (1.12 + 0.409i)23-s + (−0.109 + 0.622i)25-s + (−0.594 + 0.803i)27-s + (−0.129 + 0.735i)29-s + (0.398 + 0.144i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.640 - 0.767i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.640 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69049 + 0.791365i\)
\(L(\frac12)\) \(\approx\) \(1.69049 + 0.791365i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 - 1.16i)T \)
good5 \( 1 + (1.03 - 0.871i)T + (0.868 - 4.92i)T^{2} \)
7 \( 1 + (-4.32 + 1.57i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (0.700 + 0.587i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.939 - 5.32i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (3.81 + 6.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.825 + 1.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.39 - 1.96i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.698 - 3.96i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.21 - 0.806i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (2.81 + 4.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.570 + 3.23i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (3.36 + 2.82i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (4.81 - 1.75i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 + (-0.651 + 0.547i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (3.48 - 1.26i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.36 + 7.74i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (6.18 + 10.7i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.00 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.30 + 13.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.992 + 5.62i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-4.63 + 8.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.62 + 7.23i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.25648285710385694919589653537, −10.58861350669579135143262479520, −9.202894999853737938665517598146, −8.753409921297699657322560377166, −7.50552462369052921791991304883, −7.08960429329782001814657143778, −5.02599876625279621554766152014, −4.52328955517667235174456732581, −3.31877097861172825740401446810, −1.86899557031857920100726588917, 1.35544747348856906757521515498, 2.61904502176822129147631361332, 4.10035642176424888473650251687, 5.20636872224703422308695893506, 6.38097026287731227222774821697, 7.78070849316450360335718325763, 8.284897751940992690762361303124, 8.674861785344790465322041959601, 10.17795429249040750355911650616, 11.16192736255328592677877422817

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