Properties

Label 2-24e2-1.1-c3-0-20
Degree $2$
Conductor $576$
Sign $-1$
Analytic cond. $33.9851$
Root an. cond. $5.82967$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 20·7-s + 70·13-s + 56·19-s − 125·25-s − 308·31-s − 110·37-s − 520·43-s + 57·49-s − 182·61-s − 880·67-s + 1.19e3·73-s − 884·79-s − 1.40e3·91-s − 1.33e3·97-s − 1.82e3·103-s + 646·109-s + ⋯
L(s)  = 1  − 1.07·7-s + 1.49·13-s + 0.676·19-s − 25-s − 1.78·31-s − 0.488·37-s − 1.84·43-s + 0.166·49-s − 0.382·61-s − 1.60·67-s + 1.90·73-s − 1.25·79-s − 1.61·91-s − 1.39·97-s − 1.74·103-s + 0.567·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(33.9851\)
Root analytic conductor: \(5.82967\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 576,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good5 \( 1 + p^{3} T^{2} \)
7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 + p^{3} T^{2} \)
13 \( 1 - 70 T + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 - 56 T + p^{3} T^{2} \)
23 \( 1 + p^{3} T^{2} \)
29 \( 1 + p^{3} T^{2} \)
31 \( 1 + 308 T + p^{3} T^{2} \)
37 \( 1 + 110 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + 520 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + 182 T + p^{3} T^{2} \)
67 \( 1 + 880 T + p^{3} T^{2} \)
71 \( 1 + p^{3} T^{2} \)
73 \( 1 - 1190 T + p^{3} T^{2} \)
79 \( 1 + 884 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + 1330 T + p^{3} T^{2} \)
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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

−9.809638470798279571580764215477, −9.100336199545570600055188389828, −8.167603860032875616161123613038, −7.07831348933714242290584814039, −6.22063549463307429132747192529, −5.40351085663299133029438046366, −3.88249231317028251229726666241, −3.19503281087923466256842789439, −1.58047028685932102623093765571, 0, 1.58047028685932102623093765571, 3.19503281087923466256842789439, 3.88249231317028251229726666241, 5.40351085663299133029438046366, 6.22063549463307429132747192529, 7.07831348933714242290584814039, 8.167603860032875616161123613038, 9.100336199545570600055188389828, 9.809638470798279571580764215477

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