Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 11^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 7-s + 9-s − 4·13-s − 6·17-s + 2·19-s − 2·21-s − 3·23-s + 4·27-s − 6·29-s + 7·31-s + 4·37-s + 8·39-s − 9·41-s + 4·43-s + 9·47-s − 6·49-s + 12·51-s − 4·57-s − 6·59-s + 10·61-s + 63-s − 10·67-s + 6·69-s + 11·73-s + 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 1.45·17-s + 0.458·19-s − 0.436·21-s − 0.625·23-s + 0.769·27-s − 1.11·29-s + 1.25·31-s + 0.657·37-s + 1.28·39-s − 1.40·41-s + 0.609·43-s + 1.31·47-s − 6/7·49-s + 1.68·51-s − 0.529·57-s − 0.781·59-s + 1.28·61-s + 0.125·63-s − 1.22·67-s + 0.722·69-s + 1.28·73-s + 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(48400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{48400} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 48400,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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

−14.88954405153804, −14.33634074327212, −13.67653528948875, −13.33281390797419, −12.58073884382007, −12.11026173466606, −11.74334807429801, −11.26581195034555, −10.75828876046642, −10.33070679501555, −9.589522161285006, −9.256809753616215, −8.414722519232902, −7.972086684016415, −7.261734585229847, −6.735714139912667, −6.286026782883741, −5.545674663778140, −5.197601596992994, −4.490580011508967, −4.173660113099184, −3.096233385565348, −2.401593370748741, −1.750107095354555, −0.7195632577622027, 0, 0.7195632577622027, 1.750107095354555, 2.401593370748741, 3.096233385565348, 4.173660113099184, 4.490580011508967, 5.197601596992994, 5.545674663778140, 6.286026782883741, 6.735714139912667, 7.261734585229847, 7.972086684016415, 8.414722519232902, 9.256809753616215, 9.589522161285006, 10.33070679501555, 10.75828876046642, 11.26581195034555, 11.74334807429801, 12.11026173466606, 12.58073884382007, 13.33281390797419, 13.67653528948875, 14.33634074327212, 14.88954405153804

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