↑

Properties

Label 2-72450-1.1-c1-0-118
Degree $2$
Conductor $72450$
Sign $-1$
Analytic cond. $578.516$
Root an. cond. $24.0523$
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 − 7-s + 8-s + 6·11-s − 6·13-s − 14-s + 16-s + 6·19-s + 6·22-s + 23-s − 6·26-s − 28-s + 6·29-s + 2·31-s + 32-s + 4·37-s + 6·38-s − 6·41-s + 8·43-s + 6·44-s + 46-s − 12·47-s + 49-s − 6·52-s − 14·53-s − 56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s + 1.80·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s + 1.27·22-s + 0.208·23-s − 1.17·26-s − 0.188·28-s + 1.11·29-s + 0.359·31-s + 0.176·32-s + 0.657·37-s + 0.973·38-s − 0.937·41-s + 1.21·43-s + 0.904·44-s + 0.147·46-s − 1.75·47-s + 1/7·49-s − 0.832·52-s − 1.92·53-s − 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(72450\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(578.516\)
Root analytic conductor: \(24.0523\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 72450,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−14.36173193406767, −14.02749017736068, −13.41703041591762, −12.75742433437022, −12.31256799035864, −11.92685065937228, −11.58573355841547, −11.03563212819030, −10.23480606330477, −9.749964211324115, −9.419143608648276, −8.937287960923509, −8.000381758378244, −7.654502893528450, −6.845970269430457, −6.697973486646481, −6.070186630439349, −5.391076411252810, −4.742422742165642, −4.428345986026524, −3.683535243417457, −2.993653930134546, −2.705557500738170, −1.609024771052050, −1.160153703913900, 0, 1.160153703913900, 1.609024771052050, 2.705557500738170, 2.993653930134546, 3.683535243417457, 4.428345986026524, 4.742422742165642, 5.391076411252810, 6.070186630439349, 6.697973486646481, 6.845970269430457, 7.654502893528450, 8.000381758378244, 8.937287960923509, 9.419143608648276, 9.749964211324115, 10.23480606330477, 11.03563212819030, 11.58573355841547, 11.92685065937228, 12.31256799035864, 12.75742433437022, 13.41703041591762, 14.02749017736068, 14.36173193406767

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