Properties

Label 2-10080-1.1-c1-0-20
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 + 2·13-s + 8·17-s + 6·19-s + 4·23-s + 25-s − 2·29-s − 4·31-s + 35-s − 8·37-s + 10·41-s + 10·43-s − 10·47-s + 49-s + 2·53-s − 4·55-s − 4·59-s + 4·61-s − 2·65-s + 10·67-s + 6·71-s + 6·73-s − 4·77-s + 8·79-s − 8·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s + 1.20·11-s + 0.554·13-s + 1.94·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.169·35-s − 1.31·37-s + 1.56·41-s + 1.52·43-s − 1.45·47-s + 1/7·49-s + 0.274·53-s − 0.539·55-s − 0.520·59-s + 0.512·61-s − 0.248·65-s + 1.22·67-s + 0.712·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s − 0.878·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.474292180\)
\(L(\frac12)\) \(\approx\) \(2.474292180\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.61010813220752, −16.06611667223787, −15.64058773573859, −14.78655319163645, −14.29763835055251, −13.98494458179849, −13.10612114737931, −12.35411908538688, −12.19602493927304, −11.27104712328187, −11.03460839512113, −10.02437022941944, −9.501080277513116, −9.066036368923217, −8.239938898948910, −7.540569653205021, −7.097874485421190, −6.300081451300631, −5.578027461252717, −5.040728035866151, −3.794857469705622, −3.636699193712244, −2.801423053758577, −1.431834185284282, −0.8442548015094900, 0.8442548015094900, 1.431834185284282, 2.801423053758577, 3.636699193712244, 3.794857469705622, 5.040728035866151, 5.578027461252717, 6.300081451300631, 7.097874485421190, 7.540569653205021, 8.239938898948910, 9.066036368923217, 9.501080277513116, 10.02437022941944, 11.03460839512113, 11.27104712328187, 12.19602493927304, 12.35411908538688, 13.10612114737931, 13.98494458179849, 14.29763835055251, 14.78655319163645, 15.64058773573859, 16.06611667223787, 16.61010813220752

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