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  − 3·3-s + 6·9-s + 2·13-s + 2·17-s − 2·19-s + 23-s − 9·27-s − 29-s + 10·31-s − 8·37-s − 6·39-s + 3·41-s − 5·43-s + 8·47-s − 6·51-s − 6·53-s + 6·57-s + 2·59-s + 9·61-s + 7·67-s − 3·69-s − 6·71-s + 10·73-s + 10·79-s + 9·81-s − 9·83-s + 3·87-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s + 0.554·13-s + 0.485·17-s − 0.458·19-s + 0.208·23-s − 1.73·27-s − 0.185·29-s + 1.79·31-s − 1.31·37-s − 0.960·39-s + 0.468·41-s − 0.762·43-s + 1.16·47-s − 0.840·51-s − 0.824·53-s + 0.794·57-s + 0.260·59-s + 1.15·61-s + 0.855·67-s − 0.361·69-s − 0.712·71-s + 1.17·73-s + 1.12·79-s + 81-s − 0.987·83-s + 0.321·87-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  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.128800507$
$L(\frac12)$  $\approx$  $1.128800507$
$L(\frac{3}{2})$   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 \( 1 + p T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 9 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.91498941666788, −15.36887667923749, −14.70706598488257, −13.89960925756867, −13.45266023630313, −12.71380335725390, −12.31398857609021, −11.82675377440844, −11.25198945318335, −10.83736754307585, −10.19176499902821, −9.887822255303268, −8.977788772871937, −8.316094727841233, −7.631286796199661, −6.816691865120045, −6.504064117891819, −5.850279688410561, −5.300097470492605, −4.734586156369102, −4.066673653961266, −3.303572875341527, −2.189296861766238, −1.228816305375612, −0.5580033408592533, 0.5580033408592533, 1.228816305375612, 2.189296861766238, 3.303572875341527, 4.066673653961266, 4.734586156369102, 5.300097470492605, 5.850279688410561, 6.504064117891819, 6.816691865120045, 7.631286796199661, 8.316094727841233, 8.977788772871937, 9.887822255303268, 10.19176499902821, 10.83736754307585, 11.25198945318335, 11.82675377440844, 12.31398857609021, 12.71380335725390, 13.45266023630313, 13.89960925756867, 14.70706598488257, 15.36887667923749, 15.91498941666788

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