↑

Properties

Label 2-57330-1.1-c1-0-68
Degree $2$
Conductor $57330$
Sign $-1$
Analytic cond. $457.782$
Root an. cond. $21.3958$
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 + 4-s − 5-s − 8-s + 10-s − 13-s + 16-s − 2·19-s − 20-s − 6·23-s + 25-s + 26-s − 6·29-s + 4·31-s − 32-s + 2·37-s + 2·38-s + 40-s + 8·43-s + 6·46-s − 50-s − 52-s − 12·53-s + 6·58-s + 6·59-s + 10·61-s − 4·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.277·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 1.25·23-s + 1/5·25-s + 0.196·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.328·37-s + 0.324·38-s + 0.158·40-s + 1.21·43-s + 0.884·46-s − 0.141·50-s − 0.138·52-s − 1.64·53-s + 0.787·58-s + 0.781·59-s + 1.28·61-s − 0.508·62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(57330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(457.782\)
Root analytic conductor: \(21.3958\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 57330,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−14.66599971060393, −14.28276087696886, −13.57408564848925, −13.03204976371579, −12.46136791705564, −12.01005044891543, −11.53616064692905, −10.97104922409089, −10.55644860226734, −9.926311918698985, −9.480677720761863, −8.976209719228202, −8.299929974679109, −7.894331786474923, −7.481893857733578, −6.798191346477134, −6.260693849955980, −5.719551854692145, −5.013695964260965, −4.247146204568204, −3.814386234715375, −3.009189752344561, −2.317814509252334, −1.727646209092391, −0.7757800544022981, 0, 0.7757800544022981, 1.727646209092391, 2.317814509252334, 3.009189752344561, 3.814386234715375, 4.247146204568204, 5.013695964260965, 5.719551854692145, 6.260693849955980, 6.798191346477134, 7.481893857733578, 7.894331786474923, 8.299929974679109, 8.976209719228202, 9.480677720761863, 9.926311918698985, 10.55644860226734, 10.97104922409089, 11.53616064692905, 12.01005044891543, 12.46136791705564, 13.03204976371579, 13.57408564848925, 14.28276087696886, 14.66599971060393

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