Properties

Label 2-2880-1.1-c1-0-25
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 − 4·11-s − 2·13-s + 2·17-s + 8·19-s + 4·23-s + 25-s − 6·29-s − 2·37-s + 6·41-s + 4·43-s − 12·47-s − 7·49-s − 6·53-s + 4·55-s − 12·59-s − 14·61-s + 2·65-s − 12·67-s + 2·73-s + 8·79-s + 4·83-s − 2·85-s − 2·89-s − 8·95-s − 14·97-s − 14·101-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.328·37-s + 0.937·41-s + 0.609·43-s − 1.75·47-s − 49-s − 0.824·53-s + 0.539·55-s − 1.56·59-s − 1.79·61-s + 0.248·65-s − 1.46·67-s + 0.234·73-s + 0.900·79-s + 0.439·83-s − 0.216·85-s − 0.211·89-s − 0.820·95-s − 1.42·97-s − 1.39·101-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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−8.150684156310851114720609991661, −7.64638497790967782031689844531, −7.16490454639776840227424424985, −5.98838172699024556681992333300, −5.20439272618740772623254047156, −4.65022599724188488347085550155, −3.34351096140425392586912650668, −2.85471021679403173274520085916, −1.44764300242999129430488129246, 0, 1.44764300242999129430488129246, 2.85471021679403173274520085916, 3.34351096140425392586912650668, 4.65022599724188488347085550155, 5.20439272618740772623254047156, 5.98838172699024556681992333300, 7.16490454639776840227424424985, 7.64638497790967782031689844531, 8.150684156310851114720609991661

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