Properties

Label 2-2880-1.1-c1-0-18
Degree $2$
Conductor $2880$
Sign $1$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

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

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.424035418\)
\(L(\frac12)\) \(\approx\) \(2.424035418\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.610742879040566128162935487774, −8.094484908266087825781152713303, −7.38986169737030489086018454093, −6.52411082007979697711723996792, −5.34437678528517976086556923571, −5.16731330798328126955314301729, −4.16841981012504174266678012358, −2.94471466186386712232384138589, −2.05937366765957406711596802667, −1.01814837175580913037691321841, 1.01814837175580913037691321841, 2.05937366765957406711596802667, 2.94471466186386712232384138589, 4.16841981012504174266678012358, 5.16731330798328126955314301729, 5.34437678528517976086556923571, 6.52411082007979697711723996792, 7.38986169737030489086018454093, 8.094484908266087825781152713303, 8.610742879040566128162935487774

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