↑

Properties

Label 2-262080-1.1-c1-0-145
Degree $2$
Conductor $262080$
Sign $-1$
Analytic cond. $2092.71$
Root an. cond. $45.7462$
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  − 5-s − 7-s − 6·11-s − 13-s + 4·17-s + 4·19-s − 4·23-s + 25-s − 6·29-s + 10·31-s + 35-s − 6·37-s − 2·43-s − 4·47-s + 49-s + 6·55-s − 4·59-s + 10·61-s + 65-s + 8·67-s − 8·71-s − 2·73-s + 6·77-s − 12·83-s − 4·85-s + 91-s − 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s + 0.970·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.79·31-s + 0.169·35-s − 0.986·37-s − 0.304·43-s − 0.583·47-s + 1/7·49-s + 0.809·55-s − 0.520·59-s + 1.28·61-s + 0.124·65-s + 0.977·67-s − 0.949·71-s − 0.234·73-s + 0.683·77-s − 1.31·83-s − 0.433·85-s + 0.104·91-s − 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(262080\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(2092.71\)
Root analytic conductor: \(45.7462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 262080,\ (\ :1/2),\ -1)\)

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

−13.07685250164646, −12.54102466259572, −12.15797458617772, −11.68529292725890, −11.26716079793336, −10.68077291375473, −10.10831527486684, −9.929905266506144, −9.579906809712768, −8.636112343438257, −8.359302148557847, −7.880084181372494, −7.372467924132745, −7.182939801700206, −6.359051386704692, −5.806681634994332, −5.408026474494675, −4.928371212909624, −4.451827483859808, −3.621448528659512, −3.299015401417262, −2.732254583189745, −2.223551700222586, −1.431686132827576, −0.6208819113440364, 0, 0.6208819113440364, 1.431686132827576, 2.223551700222586, 2.732254583189745, 3.299015401417262, 3.621448528659512, 4.451827483859808, 4.928371212909624, 5.408026474494675, 5.806681634994332, 6.359051386704692, 7.182939801700206, 7.372467924132745, 7.880084181372494, 8.359302148557847, 8.636112343438257, 9.579906809712768, 9.929905266506144, 10.10831527486684, 10.68077291375473, 11.26716079793336, 11.68529292725890, 12.15797458617772, 12.54102466259572, 13.07685250164646

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