Properties

Label 2-2880-1.1-c1-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s − 2·13-s + 6·17-s − 4·19-s + 6·23-s + 25-s + 6·29-s + 4·31-s + 2·35-s − 2·37-s − 6·41-s − 10·43-s − 6·47-s − 3·49-s − 6·53-s − 12·59-s − 2·61-s + 2·65-s + 2·67-s − 12·71-s + 2·73-s − 8·79-s − 6·83-s − 6·85-s + 6·89-s + 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s − 0.554·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.338·35-s − 0.328·37-s − 0.937·41-s − 1.52·43-s − 0.875·47-s − 3/7·49-s − 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.248·65-s + 0.244·67-s − 1.42·71-s + 0.234·73-s − 0.900·79-s − 0.658·83-s − 0.650·85-s + 0.635·89-s + 0.419·91-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: $\chi_{2880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -1)\)

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 \( 1 + 2 T + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−8.354217905819492276622407670543, −7.68981058172178643196515579808, −6.79665463493597297389359578264, −6.30016858748825461633606551620, −5.17381357315305902780091527317, −4.55074829093045740786740684957, −3.35135398071564319960087881565, −2.91285425746382377562417869193, −1.41105370921345955286948063209, 0, 1.41105370921345955286948063209, 2.91285425746382377562417869193, 3.35135398071564319960087881565, 4.55074829093045740786740684957, 5.17381357315305902780091527317, 6.30016858748825461633606551620, 6.79665463493597297389359578264, 7.68981058172178643196515579808, 8.354217905819492276622407670543

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