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Properties

Label 2-8640-1.1-c1-0-81
Degree $2$
Conductor $8640$
Sign $-1$
Analytic cond. $68.9907$
Root an. cond. $8.30606$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s − 2·11-s − 4·13-s + 17-s + 5·19-s + 5·23-s + 25-s − 8·29-s + 7·31-s − 4·35-s + 6·37-s + 6·41-s + 2·43-s + 8·47-s + 9·49-s − 9·53-s − 2·55-s − 4·59-s − 13·61-s − 4·65-s + 10·67-s − 6·71-s − 6·73-s + 8·77-s + 9·79-s + 17·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s − 0.603·11-s − 1.10·13-s + 0.242·17-s + 1.14·19-s + 1.04·23-s + 1/5·25-s − 1.48·29-s + 1.25·31-s − 0.676·35-s + 0.986·37-s + 0.937·41-s + 0.304·43-s + 1.16·47-s + 9/7·49-s − 1.23·53-s − 0.269·55-s − 0.520·59-s − 1.66·61-s − 0.496·65-s + 1.22·67-s − 0.712·71-s − 0.702·73-s + 0.911·77-s + 1.01·79-s + 1.86·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8640\)    =    \(2^{6} \cdot 3^{3} \cdot 5\)
Sign: $-1$
Analytic conductor: \(68.9907\)
Root analytic conductor: \(8.30606\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8640,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 - 17 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.49021375169655068173068034665, −6.71401739160618710686470724247, −6.05954716596321482019902208949, −5.42016231751570044953010087520, −4.74472917965118669238450357257, −3.74003194570339284765478358434, −2.85015981870912461136487956707, −2.56805366284799917081919448512, −1.12737072596111806526309478815, 0, 1.12737072596111806526309478815, 2.56805366284799917081919448512, 2.85015981870912461136487956707, 3.74003194570339284765478358434, 4.74472917965118669238450357257, 5.42016231751570044953010087520, 6.05954716596321482019902208949, 6.71401739160618710686470724247, 7.49021375169655068173068034665

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