Properties

Label 2-4730-1.1-c1-0-113
Degree $2$
Conductor $4730$
Sign $-1$
Analytic cond. $37.7692$
Root an. cond. $6.14566$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s + 11-s − 3·12-s + 14-s − 3·15-s + 16-s − 3·17-s + 6·18-s − 19-s + 20-s − 3·21-s + 22-s − 4·23-s − 3·24-s + 25-s − 9·27-s + 28-s − 5·29-s − 3·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s + 0.301·11-s − 0.866·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s + 0.213·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s − 1.73·27-s + 0.188·28-s − 0.928·29-s − 0.547·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
Sign: $-1$
Analytic conductor: \(37.7692\)
Root analytic conductor: \(6.14566\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4730,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−7.61374268882365308491809010853, −6.74763403033556146146528064874, −6.46423147525257231926186318384, −5.46571368487045377522152451761, −5.30601840409615725349846331665, −4.35411509772499546560136238762, −3.71824275067621020081822687674, −2.20720314696272689339072911551, −1.41334326792664660603085285950, 0, 1.41334326792664660603085285950, 2.20720314696272689339072911551, 3.71824275067621020081822687674, 4.35411509772499546560136238762, 5.30601840409615725349846331665, 5.46571368487045377522152451761, 6.46423147525257231926186318384, 6.74763403033556146146528064874, 7.61374268882365308491809010853

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