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Properties

Label 2-432-16.5-c1-0-19
Degree $2$
Conductor $432$
Sign $-0.242 + 0.970i$
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  + (−1.20 − 0.741i)2-s + (0.899 + 1.78i)4-s + (−0.431 − 0.431i)5-s − 0.145i·7-s + (0.242 − 2.81i)8-s + (0.199 + 0.839i)10-s + (−2.42 − 2.42i)11-s + (0.201 − 0.201i)13-s + (−0.107 + 0.174i)14-s + (−2.38 + 3.21i)16-s + 3.15·17-s + (3.09 − 3.09i)19-s + (0.382 − 1.15i)20-s + (1.12 + 4.72i)22-s − 5.99i·23-s + ⋯
L(s)  = 1  + (−0.851 − 0.524i)2-s + (0.449 + 0.893i)4-s + (−0.192 − 0.192i)5-s − 0.0548i·7-s + (0.0856 − 0.996i)8-s + (0.0630 + 0.265i)10-s + (−0.731 − 0.731i)11-s + (0.0558 − 0.0558i)13-s + (−0.0287 + 0.0466i)14-s + (−0.595 + 0.803i)16-s + 0.764·17-s + (0.709 − 0.709i)19-s + (0.0855 − 0.259i)20-s + (0.239 + 1.00i)22-s − 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.242 + 0.970i$
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: \((2,\ 432,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465951 - 0.596978i\)
\(L(\frac12)\) \(\approx\) \(0.465951 - 0.596978i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 0.741i)T \)
3 \( 1 \)
good5 \( 1 + (0.431 + 0.431i)T + 5iT^{2} \)
7 \( 1 + 0.145iT - 7T^{2} \)
11 \( 1 + (2.42 + 2.42i)T + 11iT^{2} \)
13 \( 1 + (-0.201 + 0.201i)T - 13iT^{2} \)
17 \( 1 - 3.15T + 17T^{2} \)
19 \( 1 + (-3.09 + 3.09i)T - 19iT^{2} \)
23 \( 1 + 5.99iT - 23T^{2} \)
29 \( 1 + (0.395 - 0.395i)T - 29iT^{2} \)
31 \( 1 - 3.98T + 31T^{2} \)
37 \( 1 + (5.29 + 5.29i)T + 37iT^{2} \)
41 \( 1 - 1.76iT - 41T^{2} \)
43 \( 1 + (8.07 + 8.07i)T + 43iT^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 + (-7.91 - 7.91i)T + 53iT^{2} \)
59 \( 1 + (6.18 + 6.18i)T + 59iT^{2} \)
61 \( 1 + (0.789 - 0.789i)T - 61iT^{2} \)
67 \( 1 + (-6.34 + 6.34i)T - 67iT^{2} \)
71 \( 1 - 3.39iT - 71T^{2} \)
73 \( 1 + 2.23iT - 73T^{2} \)
79 \( 1 + 7.01T + 79T^{2} \)
83 \( 1 + (-4.08 + 4.08i)T - 83iT^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−10.67045719086937933267393860331, −10.18611023683680315918010323385, −9.006889746292426583972889968083, −8.311364936559904510390593712272, −7.48989059809705803013518384289, −6.38602513283620056346715749469, −5.01004532892802700604125327063, −3.59085573191437066675290018805, −2.49600753099060747652748433079, −0.65921625905011355910583566383, 1.59232036597675034128697179553, 3.22450465296906786048086097404, 4.98389687583302022355759726035, 5.81617053827835997073489406680, 7.09112595056835569552506258670, 7.68329795312956185876154967450, 8.567890259989310210711120348834, 9.829417071714389608816165612485, 10.08481907350354071234271891363, 11.33206595351793733201772237935

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