Properties

Label 2-12480-1.1-c1-0-79
Degree $2$
Conductor $12480$
Sign $-1$
Analytic cond. $99.6533$
Root an. cond. $9.98265$
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  − 3-s + 5-s + 4·7-s + 9-s + 13-s − 15-s − 2·17-s − 4·19-s − 4·21-s + 8·23-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·35-s − 2·37-s − 39-s − 6·41-s − 12·43-s + 45-s + 9·49-s + 2·51-s − 10·53-s + 4·57-s + 10·61-s + 4·63-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.160·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s + 1.28·61-s + 0.503·63-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(12480\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(99.6533\)
Root analytic conductor: \(9.98265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 12480,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−16.82676073814800, −16.12368470843964, −15.34824609825697, −14.77984469801321, −14.59955697035254, −13.71347914839238, −13.10416777557712, −12.78279041541939, −11.85248972642698, −11.36110555767048, −10.93310309311402, −10.50137196286168, −9.707572418469722, −8.875556090809650, −8.544464322370439, −7.795898843888803, −6.978996210136638, −6.613747486694067, −5.605719489211694, −5.180358394209827, −4.630030192694407, −3.868942667235526, −2.843751344603721, −1.758116976427070, −1.436564479698511, 0, 1.436564479698511, 1.758116976427070, 2.843751344603721, 3.868942667235526, 4.630030192694407, 5.180358394209827, 5.605719489211694, 6.613747486694067, 6.978996210136638, 7.795898843888803, 8.544464322370439, 8.875556090809650, 9.707572418469722, 10.50137196286168, 10.93310309311402, 11.36110555767048, 11.85248972642698, 12.78279041541939, 13.10416777557712, 13.71347914839238, 14.59955697035254, 14.77984469801321, 15.34824609825697, 16.12368470843964, 16.82676073814800

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