Properties

Label 2-2520-1.1-c1-0-16
Degree $2$
Conductor $2520$
Sign $1$
Analytic cond. $20.1223$
Root an. cond. $4.48578$
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 + 2·17-s + 4·23-s + 25-s + 2·29-s − 4·31-s + 35-s − 10·37-s − 6·41-s − 4·43-s + 8·47-s + 49-s + 6·53-s + 4·55-s + 12·59-s + 6·61-s + 2·65-s − 12·67-s + 6·73-s + 4·77-s + 12·83-s + 2·85-s − 6·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.169·35-s − 1.64·37-s − 0.937·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 1.56·59-s + 0.768·61-s + 0.248·65-s − 1.46·67-s + 0.702·73-s + 0.455·77-s + 1.31·83-s + 0.216·85-s − 0.635·89-s + 0.209·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(20.1223\)
Root analytic conductor: \(4.48578\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.385677429\)
\(L(\frac12)\) \(\approx\) \(2.385677429\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.828267909038430753578431298681, −8.389950023653329005104130022496, −7.18101803207037383757614184473, −6.72169674737378724213074355441, −5.74459909021059259985371704177, −5.09197924761052936762378065930, −4.02173853824443339466037204458, −3.27651630598498610312463507518, −1.97166250502449888467333045049, −1.06935660167784498457582059702, 1.06935660167784498457582059702, 1.97166250502449888467333045049, 3.27651630598498610312463507518, 4.02173853824443339466037204458, 5.09197924761052936762378065930, 5.74459909021059259985371704177, 6.72169674737378724213074355441, 7.18101803207037383757614184473, 8.389950023653329005104130022496, 8.828267909038430753578431298681

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