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  − 2·3-s + 9-s + 11-s + 3·13-s + 2·17-s + 5·19-s + 7·23-s + 4·27-s − 6·29-s − 4·31-s − 2·33-s + 5·37-s − 6·39-s − 5·41-s + 6·43-s − 9·47-s − 4·51-s − 11·53-s − 10·57-s − 8·59-s − 12·61-s − 4·67-s − 14·69-s + 4·71-s − 12·73-s − 14·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.485·17-s + 1.14·19-s + 1.45·23-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s + 0.821·37-s − 0.960·39-s − 0.780·41-s + 0.914·43-s − 1.31·47-s − 0.560·51-s − 1.51·53-s − 1.32·57-s − 1.04·59-s − 1.53·61-s − 0.488·67-s − 1.68·69-s + 0.474·71-s − 1.40·73-s − 1.57·79-s − 1.22·81-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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−15.98832043327487, −15.66403383023683, −14.77555131716770, −14.45138141794445, −13.71635930777833, −13.07922072281213, −12.68671820239922, −11.98277052356271, −11.41757497113384, −11.15563621154915, −10.62330679366484, −9.895555436077599, −9.217501688431073, −8.849546932908162, −7.881577679128307, −7.392978959464578, −6.719616243216955, −6.016321581234005, −5.690089757807300, −4.970579660791818, −4.441509309637514, −3.367684646502037, −3.042667490320172, −1.620072122075037, −1.065895775367562, 0, 1.065895775367562, 1.620072122075037, 3.042667490320172, 3.367684646502037, 4.441509309637514, 4.970579660791818, 5.690089757807300, 6.016321581234005, 6.719616243216955, 7.392978959464578, 7.881577679128307, 8.849546932908162, 9.217501688431073, 9.895555436077599, 10.62330679366484, 11.15563621154915, 11.41757497113384, 11.98277052356271, 12.68671820239922, 13.07922072281213, 13.71635930777833, 14.45138141794445, 14.77555131716770, 15.66403383023683, 15.98832043327487

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