Properties

Degree 2
Conductor $ 2^{4} \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·9-s − 3·11-s − 13-s + 5·17-s + 6·19-s + 5·27-s − 5·29-s − 2·31-s + 3·33-s + 4·37-s + 39-s − 2·41-s + 10·43-s − 9·47-s − 5·51-s − 6·53-s − 6·57-s + 6·59-s − 12·61-s − 2·67-s + 14·73-s − 79-s + 81-s + 12·83-s + 5·87-s + 2·93-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.904·11-s − 0.277·13-s + 1.21·17-s + 1.37·19-s + 0.962·27-s − 0.928·29-s − 0.359·31-s + 0.522·33-s + 0.657·37-s + 0.160·39-s − 0.312·41-s + 1.52·43-s − 1.31·47-s − 0.700·51-s − 0.824·53-s − 0.794·57-s + 0.781·59-s − 1.53·61-s − 0.244·67-s + 1.63·73-s − 0.112·79-s + 1/9·81-s + 1.31·83-s + 0.536·87-s + 0.207·93-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  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.188914119$
$L(\frac12)$  $\approx$  $1.188914119$
$L(\frac{3}{2})$   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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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

−15.85144641491476, −15.10758036396419, −14.57153610829508, −14.11370822684438, −13.49921533266995, −12.91351808339454, −12.21294182048575, −11.97341697151439, −11.11297688040839, −10.93208340705630, −10.12818347497175, −9.562006127920262, −9.127716330599119, −8.108925521406600, −7.799922011577505, −7.248634715160592, −6.359565414619967, −5.753690363755759, −5.266482770731575, −4.865839712133335, −3.763431747196289, −3.114410937380836, −2.506855120290363, −1.409179460102188, −0.4851981731473620, 0.4851981731473620, 1.409179460102188, 2.506855120290363, 3.114410937380836, 3.763431747196289, 4.865839712133335, 5.266482770731575, 5.753690363755759, 6.359565414619967, 7.248634715160592, 7.799922011577505, 8.108925521406600, 9.127716330599119, 9.562006127920262, 10.12818347497175, 10.93208340705630, 11.11297688040839, 11.97341697151439, 12.21294182048575, 12.91351808339454, 13.49921533266995, 14.11370822684438, 14.57153610829508, 15.10758036396419, 15.85144641491476

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