Properties

Label 2-135240-1.1-c1-0-37
Degree $2$
Conductor $135240$
Sign $1$
Analytic cond. $1079.89$
Root an. cond. $32.8617$
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  + 3-s + 5-s + 9-s + 4·11-s + 15-s − 6·17-s + 4·19-s − 23-s + 25-s + 27-s − 6·29-s − 4·31-s + 4·33-s + 2·37-s + 4·41-s + 10·43-s + 45-s + 6·47-s − 6·51-s − 2·53-s + 4·55-s + 4·57-s − 2·59-s − 2·61-s + 10·67-s − 69-s + 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.258·15-s − 1.45·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.696·33-s + 0.328·37-s + 0.624·41-s + 1.52·43-s + 0.149·45-s + 0.875·47-s − 0.840·51-s − 0.274·53-s + 0.539·55-s + 0.529·57-s − 0.260·59-s − 0.256·61-s + 1.22·67-s − 0.120·69-s + 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(32.8617\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.062919055\)
\(L(\frac12)\) \(\approx\) \(4.062919055\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.60005564009160, −13.04848550937214, −12.49280658131235, −12.14127278019446, −11.41279183692250, −11.02609027890249, −10.70352748153084, −9.811220046603601, −9.475130565216434, −9.110592713609777, −8.794249029976375, −8.085161011561097, −7.437218675001547, −7.161863118820577, −6.476312687663319, −6.069501260600512, −5.482760633990166, −4.834120403365566, −4.138684669065429, −3.861018748793424, −3.181468494635750, −2.399603733234427, −2.027110856440600, −1.311677813416558, −0.5920785091469214, 0.5920785091469214, 1.311677813416558, 2.027110856440600, 2.399603733234427, 3.181468494635750, 3.861018748793424, 4.138684669065429, 4.834120403365566, 5.482760633990166, 6.069501260600512, 6.476312687663319, 7.161863118820577, 7.437218675001547, 8.085161011561097, 8.794249029976375, 9.110592713609777, 9.475130565216434, 9.811220046603601, 10.70352748153084, 11.02609027890249, 11.41279183692250, 12.14127278019446, 12.49280658131235, 13.04848550937214, 13.60005564009160

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