Properties

Label 2-4840-1.1-c1-0-10
Degree $2$
Conductor $4840$
Sign $1$
Analytic cond. $38.6475$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 5-s + 9-s − 2·13-s + 2·15-s + 6·17-s − 2·23-s + 25-s + 4·27-s − 8·29-s + 4·31-s − 2·37-s + 4·39-s − 4·41-s − 4·43-s − 45-s + 2·47-s − 7·49-s − 12·51-s − 10·53-s + 8·61-s + 2·65-s − 2·67-s + 4·69-s + 8·71-s − 10·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.417·23-s + 1/5·25-s + 0.769·27-s − 1.48·29-s + 0.718·31-s − 0.328·37-s + 0.640·39-s − 0.624·41-s − 0.609·43-s − 0.149·45-s + 0.291·47-s − 49-s − 1.68·51-s − 1.37·53-s + 1.02·61-s + 0.248·65-s − 0.244·67-s + 0.481·69-s + 0.949·71-s − 1.17·73-s − 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(38.6475\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7520748325\)
\(L(\frac12)\) \(\approx\) \(0.7520748325\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.059239194024831863572977410298, −7.57411147730452560892196209777, −6.72203117363316151400133786142, −6.05272185399923938879263742809, −5.30701214720695859044895117047, −4.83343349888270417633269853664, −3.78880864132954337287291149835, −3.01307140760090301092571089269, −1.69719506983287969813937927815, −0.50816110465666111084228077933, 0.50816110465666111084228077933, 1.69719506983287969813937927815, 3.01307140760090301092571089269, 3.78880864132954337287291149835, 4.83343349888270417633269853664, 5.30701214720695859044895117047, 6.05272185399923938879263742809, 6.72203117363316151400133786142, 7.57411147730452560892196209777, 8.059239194024831863572977410298

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