Properties

Label 2-10080-1.1-c1-0-0
Degree $2$
Conductor $10080$
Sign $1$
Analytic cond. $80.4892$
Root an. cond. $8.97157$
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 − 7-s − 4·11-s − 6·13-s − 6·17-s + 4·19-s − 4·23-s + 25-s + 2·29-s − 8·31-s + 35-s + 6·37-s − 6·41-s − 8·43-s + 49-s − 6·53-s + 4·55-s − 4·59-s + 10·61-s + 6·65-s − 8·67-s + 12·71-s − 14·73-s + 4·77-s + 16·79-s − 12·83-s + 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.169·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 1.28·61-s + 0.744·65-s − 0.977·67-s + 1.42·71-s − 1.63·73-s + 0.455·77-s + 1.80·79-s − 1.31·83-s + 0.650·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10080\)    =    \(2^{5} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(80.4892\)
Root analytic conductor: \(8.97157\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10080} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3895372224\)
\(L(\frac12)\) \(\approx\) \(0.3895372224\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−16.52189860046360, −16.01494682771090, −15.53768814768427, −14.97670379680702, −14.44504817499682, −13.64428788481272, −13.15887139334858, −12.58899027177918, −12.04175371089562, −11.38858716901564, −10.83614339398544, −9.999212678763549, −9.747934068978059, −8.917011304205502, −8.194839245585873, −7.568583813101380, −7.115314795516845, −6.396407459669255, −5.442994507762571, −4.932296533399160, −4.275150445559724, −3.303046644792985, −2.612421185534779, −1.899875698996430, −0.2730217166958997, 0.2730217166958997, 1.899875698996430, 2.612421185534779, 3.303046644792985, 4.275150445559724, 4.932296533399160, 5.442994507762571, 6.396407459669255, 7.115314795516845, 7.568583813101380, 8.194839245585873, 8.917011304205502, 9.747934068978059, 9.999212678763549, 10.83614339398544, 11.38858716901564, 12.04175371089562, 12.58899027177918, 13.15887139334858, 13.64428788481272, 14.44504817499682, 14.97670379680702, 15.53768814768427, 16.01494682771090, 16.52189860046360

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