Properties

Label 2-8640-1.1-c1-0-127
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

−7.56883127737305415311393528032, −6.80679679813662430749338369583, −5.88504425381693499042970167636, −5.44058353575193341819534685132, −4.46721073191146950941766920681, −4.20131967478516429729407971743, −2.95056694733365579276034112981, −1.90677667970264271775360494434, −1.61725955093687237692772250139, 0, 1.61725955093687237692772250139, 1.90677667970264271775360494434, 2.95056694733365579276034112981, 4.20131967478516429729407971743, 4.46721073191146950941766920681, 5.44058353575193341819534685132, 5.88504425381693499042970167636, 6.80679679813662430749338369583, 7.56883127737305415311393528032

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