Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 7^{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  − 3-s − 2·9-s − 6·11-s + 2·13-s − 6·17-s + 8·19-s + 3·23-s + 5·27-s + 3·29-s + 2·31-s + 6·33-s − 8·37-s − 2·39-s + 3·41-s + 5·43-s + 6·51-s − 12·53-s − 8·57-s + 61-s − 7·67-s − 3·69-s − 10·73-s + 4·79-s + 81-s − 3·83-s − 3·87-s + 3·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 1.80·11-s + 0.554·13-s − 1.45·17-s + 1.83·19-s + 0.625·23-s + 0.962·27-s + 0.557·29-s + 0.359·31-s + 1.04·33-s − 1.31·37-s − 0.320·39-s + 0.468·41-s + 0.762·43-s + 0.840·51-s − 1.64·53-s − 1.05·57-s + 0.128·61-s − 0.855·67-s − 0.361·69-s − 1.17·73-s + 0.450·79-s + 1/9·81-s − 0.329·83-s − 0.321·87-s + 0.317·89-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  =  1
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 \( 1 + T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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

−15.96793841437959, −15.64690026263608, −15.02650659365112, −14.13988592277478, −13.73518992646548, −13.30528600227716, −12.63864142379840, −12.13668571681092, −11.30945791957657, −11.13747911755331, −10.52072606649344, −9.997974295336216, −9.156678522446620, −8.675645425977525, −8.045416645060497, −7.425536746940333, −6.832536413100543, −6.029938520981934, −5.567650188804533, −4.945639462656363, −4.504291378310743, −3.212009778679754, −2.940449792485784, −2.028870947452467, −0.8787372296875277, 0, 0.8787372296875277, 2.028870947452467, 2.940449792485784, 3.212009778679754, 4.504291378310743, 4.945639462656363, 5.567650188804533, 6.029938520981934, 6.832536413100543, 7.425536746940333, 8.045416645060497, 8.675645425977525, 9.156678522446620, 9.997974295336216, 10.52072606649344, 11.13747911755331, 11.30945791957657, 12.13668571681092, 12.63864142379840, 13.30528600227716, 13.73518992646548, 14.13988592277478, 15.02650659365112, 15.64690026263608, 15.96793841437959

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