Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·9-s − 3·11-s + 13-s − 5·17-s − 8·19-s + 2·23-s − 9·27-s − 29-s − 2·31-s + 9·33-s − 10·37-s − 3·39-s + 6·41-s − 4·43-s − 11·47-s + 15·51-s − 6·53-s + 24·57-s − 10·59-s − 10·67-s − 6·69-s − 10·73-s + 7·79-s + 9·81-s − 12·83-s + 3·87-s + ⋯
L(s)  = 1  − 1.73·3-s + 2·9-s − 0.904·11-s + 0.277·13-s − 1.21·17-s − 1.83·19-s + 0.417·23-s − 1.73·27-s − 0.185·29-s − 0.359·31-s + 1.56·33-s − 1.64·37-s − 0.480·39-s + 0.937·41-s − 0.609·43-s − 1.60·47-s + 2.10·51-s − 0.824·53-s + 3.17·57-s − 1.30·59-s − 1.22·67-s − 0.722·69-s − 1.17·73-s + 0.787·79-s + 81-s − 1.31·83-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  =  2
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 \( 1 + p T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−16.19955833687238, −15.85472126344488, −15.29938361550611, −14.84087155847345, −13.92839396057588, −13.19292894036233, −12.84098309867898, −12.50182206101727, −11.69367244143312, −11.19941440178658, −10.73545773767571, −10.47122152800805, −9.773803711156902, −8.887736477439156, −8.439864332920884, −7.527453718629708, −6.963963147741539, −6.302081825493162, −6.041161760432025, −5.168286365778785, −4.713096704598648, −4.212294203816087, −3.208578234221576, −2.141980072574886, −1.454621328926060, 0, 0, 1.454621328926060, 2.141980072574886, 3.208578234221576, 4.212294203816087, 4.713096704598648, 5.168286365778785, 6.041161760432025, 6.302081825493162, 6.963963147741539, 7.527453718629708, 8.439864332920884, 8.887736477439156, 9.773803711156902, 10.47122152800805, 10.73545773767571, 11.19941440178658, 11.69367244143312, 12.50182206101727, 12.84098309867898, 13.19292894036233, 13.92839396057588, 14.84087155847345, 15.29938361550611, 15.85472126344488, 16.19955833687238

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