Properties

Label 2-70560-1.1-c1-0-37
Degree $2$
Conductor $70560$
Sign $1$
Analytic cond. $563.424$
Root an. cond. $23.7365$
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 − 4·23-s + 25-s + 2·29-s + 8·31-s + 6·37-s + 6·41-s − 8·43-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 + 16·79-s + 12·83-s + 6·85-s + 14·89-s − 4·95-s − 18·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-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 + 0.986·37-s + 0.937·41-s − 1.21·43-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 + 1.80·79-s + 1.31·83-s + 0.650·85-s + 1.48·89-s − 0.410·95-s − 1.82·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(70560\)    =    \(2^{5} \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(563.424\)
Root analytic conductor: \(23.7365\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{70560} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 70560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.985166118\)
\(L(\frac12)\) \(\approx\) \(2.985166118\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−13.88686635419702, −13.72293775460763, −13.21074243885020, −12.64247884068954, −12.24675123654479, −11.61430521745947, −11.00048310221905, −10.50669377595416, −10.26399941381754, −9.578129361084148, −9.124680978363841, −8.273636282788650, −8.023012824303812, −7.749247787994286, −6.619203490202803, −6.290017827121205, −5.881671366798073, −5.193976037924953, −4.701999842618310, −3.905605307087660, −3.378867077048548, −2.706205318815616, −2.083670048848307, −1.264273765023744, −0.6152661795845486, 0.6152661795845486, 1.264273765023744, 2.083670048848307, 2.706205318815616, 3.378867077048548, 3.905605307087660, 4.701999842618310, 5.193976037924953, 5.881671366798073, 6.290017827121205, 6.619203490202803, 7.749247787994286, 8.023012824303812, 8.273636282788650, 9.124680978363841, 9.578129361084148, 10.26399941381754, 10.50669377595416, 11.00048310221905, 11.61430521745947, 12.24675123654479, 12.64247884068954, 13.21074243885020, 13.72293775460763, 13.88686635419702

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