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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s − 4·11-s + 2·13-s + 2·19-s − 4·23-s + 4·27-s + 10·29-s + 4·31-s + 8·33-s + 2·37-s − 4·39-s + 12·41-s − 4·43-s − 4·47-s − 2·53-s − 4·57-s + 10·59-s − 6·61-s + 4·67-s + 8·69-s + 12·71-s − 4·73-s + 4·79-s − 11·81-s − 14·83-s − 20·87-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.458·19-s − 0.834·23-s + 0.769·27-s + 1.85·29-s + 0.718·31-s + 1.39·33-s + 0.328·37-s − 0.640·39-s + 1.87·41-s − 0.609·43-s − 0.583·47-s − 0.274·53-s − 0.529·57-s + 1.30·59-s − 0.768·61-s + 0.488·67-s + 0.963·69-s + 1.42·71-s − 0.468·73-s + 0.450·79-s − 1.22·81-s − 1.53·83-s − 2.14·87-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  =  0
Selberg data  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.052181959$
$L(\frac12)$  $\approx$  $1.052181959$
$L(\frac{3}{2})$   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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.80374054424484, −15.44124623971999, −14.54558865043013, −13.93708249659122, −13.56911593983040, −12.69712508781895, −12.46125481999016, −11.75472473891157, −11.25079427374550, −10.82682713116824, −10.12847529745513, −9.879837234909663, −8.915132055420005, −8.104362665540047, −7.970990142175662, −6.894094209463480, −6.466579224390080, −5.769935936436316, −5.368679744330323, −4.675515163975137, −4.087329980244214, −2.995114853896317, −2.503703936317561, −1.275560326830873, −0.4981163886187724, 0.4981163886187724, 1.275560326830873, 2.503703936317561, 2.995114853896317, 4.087329980244214, 4.675515163975137, 5.368679744330323, 5.769935936436316, 6.466579224390080, 6.894094209463480, 7.970990142175662, 8.104362665540047, 8.915132055420005, 9.879837234909663, 10.12847529745513, 10.82682713116824, 11.25079427374550, 11.75472473891157, 12.46125481999016, 12.69712508781895, 13.56911593983040, 13.93708249659122, 14.54558865043013, 15.44124623971999, 15.80374054424484

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