↑

Properties

Label 2-101430-1.1-c1-0-8
Degree $2$
Conductor $101430$
Sign $1$
Analytic cond. $809.922$
Root an. cond. $28.4591$
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-s + 4-s − 5-s + 8-s − 10-s − 4·11-s − 2·13-s + 16-s − 6·17-s + 4·19-s − 20-s − 4·22-s − 23-s + 25-s − 2·26-s − 2·29-s + 8·31-s + 32-s − 6·34-s + 6·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s − 4·44-s − 46-s + 50-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 1.20·11-s − 0.554·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s − 0.208·23-s + 1/5·25-s − 0.392·26-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s + 0.986·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.603·44-s − 0.147·46-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(101430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(809.922\)
Root analytic conductor: \(28.4591\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 101430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.922107103\)
\(L(\frac12)\) \(\approx\) \(1.922107103\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
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

−13.61556596929999, −13.36364020692464, −12.78500178261784, −12.36501760645066, −11.79128328425774, −11.28048931875678, −11.09555385175029, −10.19563326715438, −10.03079382676724, −9.378487080159800, −8.552547757522906, −8.248799407210301, −7.650492873294445, −7.147961694767824, −6.699013071850985, −6.088452654023174, −5.414195556619556, −4.966508060755879, −4.532223276568400, −3.924272544443326, −3.271064299818045, −2.538595466467886, −2.356844921115286, −1.322165472757181, −0.3795757003017342, 0.3795757003017342, 1.322165472757181, 2.356844921115286, 2.538595466467886, 3.271064299818045, 3.924272544443326, 4.532223276568400, 4.966508060755879, 5.414195556619556, 6.088452654023174, 6.699013071850985, 7.147961694767824, 7.650492873294445, 8.248799407210301, 8.552547757522906, 9.378487080159800, 10.03079382676724, 10.19563326715438, 11.09555385175029, 11.28048931875678, 11.79128328425774, 12.36501760645066, 12.78500178261784, 13.36364020692464, 13.61556596929999

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