Properties

Degree $2$
Conductor $2880$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 2·13-s − 6·17-s + 4·19-s + 25-s − 6·29-s + 8·31-s + 4·35-s − 2·37-s + 6·41-s + 4·43-s + 9·49-s − 6·53-s + 10·61-s + 2·65-s + 4·67-s + 2·73-s + 8·79-s + 12·83-s + 6·85-s − 18·89-s + 8·91-s − 4·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 9/7·49-s − 0.824·53-s + 1.28·61-s + 0.248·65-s + 0.488·67-s + 0.234·73-s + 0.900·79-s + 1.31·83-s + 0.650·85-s − 1.90·89-s + 0.838·91-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-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

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{2880} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9458231982\)
\(L(\frac12)\) \(\approx\) \(0.9458231982\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−18.95084820717348, −18.05024487223851, −17.45496776752833, −16.73305726175640, −16.04379862041068, −15.66623132427988, −15.10650071676467, −14.15737590779839, −13.51727773855999, −12.87117787424069, −12.38858721562320, −11.56779429771176, −10.95877204552376, −10.04396027939565, −9.499765001323408, −8.924029029653426, −7.960977050917164, −7.151136075388365, −6.598998000096462, −5.829318607508473, −4.809232432910944, −3.946282808448207, −3.128869235469911, −2.288742993655312, −0.5687648098120035, 0.5687648098120035, 2.288742993655312, 3.128869235469911, 3.946282808448207, 4.809232432910944, 5.829318607508473, 6.598998000096462, 7.151136075388365, 7.960977050917164, 8.924029029653426, 9.499765001323408, 10.04396027939565, 10.95877204552376, 11.56779429771176, 12.38858721562320, 12.87117787424069, 13.51727773855999, 14.15737590779839, 15.10650071676467, 15.66623132427988, 16.04379862041068, 16.73305726175640, 17.45496776752833, 18.05024487223851, 18.95084820717348

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