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Properties

Label 2-432-48.35-c1-0-17
Degree $2$
Conductor $432$
Sign $0.831 + 0.555i$
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  + (−0.604 + 1.27i)2-s + (−1.26 − 1.54i)4-s + (0.217 + 0.217i)5-s − 3.81·7-s + (2.74 − 0.688i)8-s + (−0.409 + 0.146i)10-s + (3.83 − 3.83i)11-s + (−2.88 − 2.88i)13-s + (2.30 − 4.87i)14-s + (−0.778 + 3.92i)16-s + 4.71i·17-s + (4.77 − 4.77i)19-s + (0.0601 − 0.611i)20-s + (2.58 + 7.21i)22-s − 6.18i·23-s + ⋯
L(s)  = 1  + (−0.427 + 0.904i)2-s + (−0.634 − 0.772i)4-s + (0.0972 + 0.0972i)5-s − 1.44·7-s + (0.969 − 0.243i)8-s + (−0.129 + 0.0463i)10-s + (1.15 − 1.15i)11-s + (−0.801 − 0.801i)13-s + (0.616 − 1.30i)14-s + (−0.194 + 0.980i)16-s + 1.14i·17-s + (1.09 − 1.09i)19-s + (0.0134 − 0.136i)20-s + (0.550 + 1.53i)22-s − 1.28i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.831 + 0.555i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.831 + 0.555i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.727512 - 0.220472i\)
\(L(\frac12)\) \(\approx\) \(0.727512 - 0.220472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.604 - 1.27i)T \)
3 \( 1 \)
good5 \( 1 + (-0.217 - 0.217i)T + 5iT^{2} \)
7 \( 1 + 3.81T + 7T^{2} \)
11 \( 1 + (-3.83 + 3.83i)T - 11iT^{2} \)
13 \( 1 + (2.88 + 2.88i)T + 13iT^{2} \)
17 \( 1 - 4.71iT - 17T^{2} \)
19 \( 1 + (-4.77 + 4.77i)T - 19iT^{2} \)
23 \( 1 + 6.18iT - 23T^{2} \)
29 \( 1 + (-0.322 + 0.322i)T - 29iT^{2} \)
31 \( 1 + 5.20iT - 31T^{2} \)
37 \( 1 + (2.92 - 2.92i)T - 37iT^{2} \)
41 \( 1 - 1.31T + 41T^{2} \)
43 \( 1 + (-0.505 - 0.505i)T + 43iT^{2} \)
47 \( 1 - 5.41T + 47T^{2} \)
53 \( 1 + (-1.85 - 1.85i)T + 53iT^{2} \)
59 \( 1 + (4.77 - 4.77i)T - 59iT^{2} \)
61 \( 1 + (0.832 + 0.832i)T + 61iT^{2} \)
67 \( 1 + (6.48 - 6.48i)T - 67iT^{2} \)
71 \( 1 - 2.47iT - 71T^{2} \)
73 \( 1 + 6.68iT - 73T^{2} \)
79 \( 1 + 9.28iT - 79T^{2} \)
83 \( 1 + (8.24 + 8.24i)T + 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 2.81T + 97T^{2} \)
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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

−10.71040812808861729402143497658, −9.981499201412678306110021973258, −9.164238151527106224608761217970, −8.419290910505877733180641772999, −7.22221270291281238385811724388, −6.34684845414007473547786577784, −5.80341797517202523800835097261, −4.29508496792451160960079195199, −2.99149493316966176294443344179, −0.58522753551584778812946905243, 1.59199966537193889519630902226, 3.08874932517599129331061896275, 4.02430379912099415547131509617, 5.31780039441312316103942440325, 6.94041066253755593726135668355, 7.40249255993810438429994419534, 9.120985560042296518998457057835, 9.553782626489777051379943223966, 9.958446895245113778789276027985, 11.38184204305662607788707450048

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