Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4·7-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s − 4·35-s − 6·37-s + 6·41-s + 8·43-s − 4·47-s + 9·49-s + 6·53-s + 4·55-s − 4·59-s + 2·61-s + 2·65-s − 8·67-s − 6·73-s − 16·77-s − 16·83-s − 2·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.583·47-s + 9/7·49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 0.702·73-s − 1.82·77-s − 1.75·83-s − 0.216·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{2880} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 2880,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.06979318790354, −18.47708112477033, −17.66639547369402, −17.14494461144065, −16.35509595895911, −16.10455672109802, −15.23817261826515, −14.51480772769296, −13.89955418364751, −13.16435221690407, −12.70730857965115, −12.05675955954907, −11.15375902164888, −10.49344969610290, −9.755924460828300, −9.083944829807577, −8.751357480986823, −7.483368351658421, −6.680922108703084, −6.215534309394891, −5.576652285983116, −4.153850364556552, −3.717759938969110, −2.625542438933622, −1.548481902619357, 0, 1.548481902619357, 2.625542438933622, 3.717759938969110, 4.153850364556552, 5.576652285983116, 6.215534309394891, 6.680922108703084, 7.483368351658421, 8.751357480986823, 9.083944829807577, 9.755924460828300, 10.49344969610290, 11.15375902164888, 12.05675955954907, 12.70730857965115, 13.16435221690407, 13.89955418364751, 14.51480772769296, 15.23817261826515, 16.10455672109802, 16.35509595895911, 17.14494461144065, 17.66639547369402, 18.47708112477033, 19.06979318790354

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