Properties

Label 2-2880-1.1-c1-0-12
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 + 4·11-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.073075474\)
\(L(\frac12)\) \(\approx\) \(2.073075474\)
\(L(\frac{3}{2})\) 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

−8.979673560126252996904408641448, −7.85971332456036455091052470017, −7.36860587905514204016946354425, −6.38799072784883331024834438481, −5.83633635666208002119272737589, −4.81681980421746127667781868965, −4.14224381134792067626093257234, −3.00246618464052305255387261031, −2.12011231568422406855465196308, −0.910745729015585711280898694007, 0.910745729015585711280898694007, 2.12011231568422406855465196308, 3.00246618464052305255387261031, 4.14224381134792067626093257234, 4.81681980421746127667781868965, 5.83633635666208002119272737589, 6.38799072784883331024834438481, 7.36860587905514204016946354425, 7.85971332456036455091052470017, 8.979673560126252996904408641448

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