Properties

Label 2-2880-1.1-c1-0-34
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 − 6·13-s + 6·17-s + 4·19-s + 25-s − 2·29-s − 8·31-s + 2·37-s + 6·41-s − 12·43-s − 8·47-s − 7·49-s + 6·53-s − 4·55-s + 12·59-s − 14·61-s − 6·65-s − 4·67-s − 8·71-s − 6·73-s − 8·79-s − 12·83-s + 6·85-s − 10·89-s + 4·95-s + 2·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s − 1.82·43-s − 1.16·47-s − 49-s + 0.824·53-s − 0.539·55-s + 1.56·59-s − 1.79·61-s − 0.744·65-s − 0.488·67-s − 0.949·71-s − 0.702·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s − 1.05·89-s + 0.410·95-s + 0.203·97-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 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−8.281438123975201335840704816115, −7.49319629717735131747504234927, −7.19464749200877042506852879968, −5.86215101035639287486075524635, −5.32415764508171872286866957381, −4.72041309651433423591886250489, −3.35030464779548994692224119619, −2.67630125421591650399057166027, −1.59198724080807310872956999404, 0, 1.59198724080807310872956999404, 2.67630125421591650399057166027, 3.35030464779548994692224119619, 4.72041309651433423591886250489, 5.32415764508171872286866957381, 5.86215101035639287486075524635, 7.19464749200877042506852879968, 7.49319629717735131747504234927, 8.281438123975201335840704816115

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