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-s − 2·9-s − 2·13-s − 2·17-s + 6·19-s + 3·23-s + 5·27-s + 7·29-s + 2·31-s − 8·37-s + 2·39-s − 5·41-s − 7·43-s + 2·51-s + 6·53-s − 6·57-s + 10·59-s − 7·61-s + 5·67-s − 3·69-s + 2·71-s + 6·73-s + 2·79-s + 81-s − 11·83-s − 7·87-s − 9·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.554·13-s − 0.485·17-s + 1.37·19-s + 0.625·23-s + 0.962·27-s + 1.29·29-s + 0.359·31-s − 1.31·37-s + 0.320·39-s − 0.780·41-s − 1.06·43-s + 0.280·51-s + 0.824·53-s − 0.794·57-s + 1.30·59-s − 0.896·61-s + 0.610·67-s − 0.361·69-s + 0.237·71-s + 0.702·73-s + 0.225·79-s + 1/9·81-s − 1.20·83-s − 0.750·87-s − 0.953·89-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  =  $(2,\ 19600,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.261059497$
$L(\frac12)$  $\approx$  $1.261059497$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−15.67421278684129, −15.28655941163759, −14.51226453389535, −14.04670321774432, −13.58257515200698, −12.94794474210604, −12.10063244315382, −11.93455396911104, −11.37410459666198, −10.73683210974377, −10.14243804477626, −9.682316719594089, −8.815409621549365, −8.489460418119914, −7.730556739510818, −6.906823098196965, −6.660642560666967, −5.785338889252186, −5.059266417691107, −4.949451636842076, −3.832123335066150, −3.073359177277145, −2.504282352171594, −1.409752310848563, −0.5009573572133848, 0.5009573572133848, 1.409752310848563, 2.504282352171594, 3.073359177277145, 3.832123335066150, 4.949451636842076, 5.059266417691107, 5.785338889252186, 6.660642560666967, 6.906823098196965, 7.730556739510818, 8.489460418119914, 8.815409621549365, 9.682316719594089, 10.14243804477626, 10.73683210974377, 11.37410459666198, 11.93455396911104, 12.10063244315382, 12.94794474210604, 13.58257515200698, 14.04670321774432, 14.51226453389535, 15.28655941163759, 15.67421278684129

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