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Properties

Label 2-432-16.5-c1-0-4
Degree $2$
Conductor $432$
Sign $0.885 - 0.465i$
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  + (−1.41 + 0.0323i)2-s + (1.99 − 0.0915i)4-s + (−1.82 − 1.82i)5-s + 1.81i·7-s + (−2.82 + 0.194i)8-s + (2.64 + 2.52i)10-s + (2.37 + 2.37i)11-s + (−3.20 + 3.20i)13-s + (−0.0588 − 2.57i)14-s + (3.98 − 0.365i)16-s + 5.42·17-s + (4.84 − 4.84i)19-s + (−3.81 − 3.48i)20-s + (−3.43 − 3.27i)22-s + 4.85i·23-s + ⋯
L(s)  = 1  + (−0.999 + 0.0228i)2-s + (0.998 − 0.0457i)4-s + (−0.817 − 0.817i)5-s + 0.687i·7-s + (−0.997 + 0.0686i)8-s + (0.835 + 0.798i)10-s + (0.715 + 0.715i)11-s + (−0.888 + 0.888i)13-s + (−0.0157 − 0.687i)14-s + (0.995 − 0.0914i)16-s + 1.31·17-s + (1.11 − 1.11i)19-s + (−0.853 − 0.779i)20-s + (−0.731 − 0.698i)22-s + 1.01i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.885 - 0.465i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.885 - 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.761045 + 0.187969i\)
\(L(\frac12)\) \(\approx\) \(0.761045 + 0.187969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0323i)T \)
3 \( 1 \)
good5 \( 1 + (1.82 + 1.82i)T + 5iT^{2} \)
7 \( 1 - 1.81iT - 7T^{2} \)
11 \( 1 + (-2.37 - 2.37i)T + 11iT^{2} \)
13 \( 1 + (3.20 - 3.20i)T - 13iT^{2} \)
17 \( 1 - 5.42T + 17T^{2} \)
19 \( 1 + (-4.84 + 4.84i)T - 19iT^{2} \)
23 \( 1 - 4.85iT - 23T^{2} \)
29 \( 1 + (-0.399 + 0.399i)T - 29iT^{2} \)
31 \( 1 - 6.64T + 31T^{2} \)
37 \( 1 + (-3.83 - 3.83i)T + 37iT^{2} \)
41 \( 1 - 7.62iT - 41T^{2} \)
43 \( 1 + (4.23 + 4.23i)T + 43iT^{2} \)
47 \( 1 - 7.76T + 47T^{2} \)
53 \( 1 + (-4.70 - 4.70i)T + 53iT^{2} \)
59 \( 1 + (3.25 + 3.25i)T + 59iT^{2} \)
61 \( 1 + (5.12 - 5.12i)T - 61iT^{2} \)
67 \( 1 + (1.80 - 1.80i)T - 67iT^{2} \)
71 \( 1 - 5.84iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + 4.10T + 79T^{2} \)
83 \( 1 + (-8.68 + 8.68i)T - 83iT^{2} \)
89 \( 1 + 5.34iT - 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−11.52380330806342715375834893763, −9.904619833008739275135570008302, −9.435115651003733753554435532450, −8.591185225915241263450156446302, −7.63840589659446128551322342552, −6.94310224819186028200169929923, −5.56073862862847888563250178142, −4.41830144467716330388133236377, −2.83342792590547559061392644481, −1.22231538862391569530370810221, 0.846407514716596379048239144905, 2.92455399060941903826559453259, 3.74860191286481363210900123604, 5.62863432534387549686990685914, 6.75070127401390170555492296312, 7.66706224831307863163256218198, 8.030641910706758022851031424269, 9.404794493241000132904440643703, 10.32564705718622231068338078909, 10.76472146913824693733353931478

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