Properties

Label 2-2880-1.1-c1-0-26
Degree $2$
Conductor $2880$
Sign $-1$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 4.47·7-s + 4.47·11-s − 4·13-s − 2·17-s + 8.94·23-s + 25-s − 6·29-s + 8.94·31-s − 4.47·35-s − 8·37-s − 8·41-s − 8.94·47-s + 13.0·49-s − 6·53-s + 4.47·55-s + 4.47·59-s − 10·61-s − 4·65-s − 8.94·67-s − 8.94·71-s + 6·73-s − 20.0·77-s − 8.94·79-s − 8.94·83-s − 2·85-s − 4·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.69·7-s + 1.34·11-s − 1.10·13-s − 0.485·17-s + 1.86·23-s + 0.200·25-s − 1.11·29-s + 1.60·31-s − 0.755·35-s − 1.31·37-s − 1.24·41-s − 1.30·47-s + 1.85·49-s − 0.824·53-s + 0.603·55-s + 0.582·59-s − 1.28·61-s − 0.496·65-s − 1.09·67-s − 1.06·71-s + 0.702·73-s − 2.27·77-s − 1.00·79-s − 0.981·83-s − 0.216·85-s − 0.423·89-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 4.47T + 7T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8.94T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.736672834619132221134147527274, −7.37118007557131887298768364978, −6.69287551516549112391053146887, −6.41133730696608170877034931075, −5.33033869973738081391347296256, −4.46560681954322177437005313439, −3.38371142387337403879635197901, −2.80789565732270295599809557896, −1.49670449859451459272590980544, 0, 1.49670449859451459272590980544, 2.80789565732270295599809557896, 3.38371142387337403879635197901, 4.46560681954322177437005313439, 5.33033869973738081391347296256, 6.41133730696608170877034931075, 6.69287551516549112391053146887, 7.37118007557131887298768364978, 8.736672834619132221134147527274

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