Properties

Label 2-432-108.11-c1-0-4
Degree $2$
Conductor $432$
Sign $-0.982 - 0.188i$
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  + (−0.0760 + 1.73i)3-s + (−1.38 + 3.79i)5-s + (−1.08 − 0.192i)7-s + (−2.98 − 0.263i)9-s + (5.69 − 2.07i)11-s + (−2.30 + 1.93i)13-s + (−6.46 − 2.68i)15-s + (−2.73 + 1.57i)17-s + (−3.28 − 1.89i)19-s + (0.415 − 1.87i)21-s + (0.347 + 1.96i)23-s + (−8.69 − 7.29i)25-s + (0.683 − 5.15i)27-s + (2.14 − 2.56i)29-s + (−3.05 + 0.538i)31-s + ⋯
L(s)  = 1  + (−0.0439 + 0.999i)3-s + (−0.618 + 1.69i)5-s + (−0.411 − 0.0725i)7-s + (−0.996 − 0.0877i)9-s + (1.71 − 0.624i)11-s + (−0.639 + 0.536i)13-s + (−1.67 − 0.692i)15-s + (−0.663 + 0.382i)17-s + (−0.753 − 0.435i)19-s + (0.0906 − 0.408i)21-s + (0.0723 + 0.410i)23-s + (−1.73 − 1.45i)25-s + (0.131 − 0.991i)27-s + (0.399 − 0.475i)29-s + (−0.548 + 0.0967i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.982 - 0.188i$
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: \((2,\ 432,\ (\ :1/2),\ -0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0849993 + 0.892015i\)
\(L(\frac12)\) \(\approx\) \(0.0849993 + 0.892015i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0760 - 1.73i)T \)
good5 \( 1 + (1.38 - 3.79i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.08 + 0.192i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-5.69 + 2.07i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (2.30 - 1.93i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (2.73 - 1.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.28 + 1.89i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.347 - 1.96i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.14 + 2.56i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.05 - 0.538i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-4.96 - 8.60i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.58 - 6.66i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-1.77 - 4.86i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.132 + 0.753i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 3.19iT - 53T^{2} \)
59 \( 1 + (7.48 + 2.72i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.13 - 6.45i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-3.17 - 3.78i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-4.30 - 7.45i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.12 + 7.14i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.29 + 5.11i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-9.12 - 7.65i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-10.9 - 6.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.13 - 2.23i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.33049801586429935351886177608, −10.84247889118151782442371863704, −9.803286509144479530183205346802, −9.121136901349513025104289611420, −7.926754598415336472305023745895, −6.51111100491751797813479354809, −6.39300757966573605894076332023, −4.42598581282222759498105050564, −3.68688809096466986169703195595, −2.70994824455377224062808324787, 0.56946666388796811530359699386, 1.95699834398419557649999991584, 3.87377018490582468122826368527, 4.87430516372036808108085029339, 6.06229439070826279716356699814, 7.08397154256056998729883204121, 7.978387511193795883419254505010, 8.976817317862257782861379654408, 9.345697215830444228573140362679, 11.01804095276527226999050173134

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