Properties

Label 2-80e2-1.1-c1-0-62
Degree $2$
Conductor $6400$
Sign $1$
Analytic cond. $51.1042$
Root an. cond. $7.14872$
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 + 4·7-s − 2·9-s + 3·11-s − 17-s + 7·19-s + 4·21-s + 4·23-s − 5·27-s − 8·29-s + 4·31-s + 3·33-s − 4·37-s − 3·41-s + 8·43-s + 9·49-s − 51-s + 12·53-s + 7·57-s + 8·59-s + 4·61-s − 8·63-s − 9·67-s + 4·69-s + 16·71-s − 11·73-s + 12·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s − 2/3·9-s + 0.904·11-s − 0.242·17-s + 1.60·19-s + 0.872·21-s + 0.834·23-s − 0.962·27-s − 1.48·29-s + 0.718·31-s + 0.522·33-s − 0.657·37-s − 0.468·41-s + 1.21·43-s + 9/7·49-s − 0.140·51-s + 1.64·53-s + 0.927·57-s + 1.04·59-s + 0.512·61-s − 1.00·63-s − 1.09·67-s + 0.481·69-s + 1.89·71-s − 1.28·73-s + 1.36·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(51.1042\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.260298056\)
\(L(\frac12)\) \(\approx\) \(3.260298056\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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

−8.061632372862743115667091017793, −7.43100129039840738482630785534, −6.83694345392637197451575922422, −5.59490587668282905141289002019, −5.33686413198391101661533754339, −4.32636959483255040005342843011, −3.62374274040571084531264233335, −2.72662471884265745613618932294, −1.83224079601896310337987946848, −0.973725220887259086889476267910, 0.973725220887259086889476267910, 1.83224079601896310337987946848, 2.72662471884265745613618932294, 3.62374274040571084531264233335, 4.32636959483255040005342843011, 5.33686413198391101661533754339, 5.59490587668282905141289002019, 6.83694345392637197451575922422, 7.43100129039840738482630785534, 8.061632372862743115667091017793

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