Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 2·13-s + 3·17-s − 8·19-s − 9·23-s + 4·27-s − 6·29-s − 5·31-s − 8·37-s + 4·39-s − 3·41-s − 10·43-s − 3·47-s − 6·51-s − 6·53-s + 16·57-s − 12·59-s − 4·61-s + 2·67-s + 18·69-s + 9·71-s + 10·73-s − 5·79-s − 11·81-s − 6·83-s + 12·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 0.554·13-s + 0.727·17-s − 1.83·19-s − 1.87·23-s + 0.769·27-s − 1.11·29-s − 0.898·31-s − 1.31·37-s + 0.640·39-s − 0.468·41-s − 1.52·43-s − 0.437·47-s − 0.840·51-s − 0.824·53-s + 2.11·57-s − 1.56·59-s − 0.512·61-s + 0.244·67-s + 2.16·69-s + 1.06·71-s + 1.17·73-s − 0.562·79-s − 1.22·81-s − 0.658·83-s + 1.28·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{19600} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.53687655072599, −15.66150420347259, −15.30010358077327, −14.45988179222699, −14.26179493267614, −13.40796893168245, −12.72756378159472, −12.31700809817828, −11.91449872075831, −11.25952858593843, −10.73576973837786, −10.26094815382063, −9.728580104587408, −8.979517399041202, −8.236830396804327, −7.806371341979523, −6.918203972259139, −6.430745841979461, −5.874957666259100, −5.288684585712835, −4.723012677283299, −3.944592739860373, −3.288267580480498, −2.130578757423733, −1.602131083981474, 0, 0, 1.602131083981474, 2.130578757423733, 3.288267580480498, 3.944592739860373, 4.723012677283299, 5.288684585712835, 5.874957666259100, 6.430745841979461, 6.918203972259139, 7.806371341979523, 8.236830396804327, 8.979517399041202, 9.728580104587408, 10.26094815382063, 10.73576973837786, 11.25952858593843, 11.91449872075831, 12.31700809817828, 12.72756378159472, 13.40796893168245, 14.26179493267614, 14.45988179222699, 15.30010358077327, 15.66150420347259, 16.53687655072599

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