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  − 2·3-s + 9-s − 4·11-s − 2·13-s − 3·17-s − 3·23-s + 4·27-s − 6·29-s − 9·31-s + 8·33-s + 4·39-s + 5·41-s + 6·43-s − 9·47-s + 6·51-s − 6·53-s − 8·59-s + 8·61-s − 14·67-s + 6·69-s − 11·71-s − 2·73-s − 9·79-s − 11·81-s + 6·83-s + 12·87-s + 11·89-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.727·17-s − 0.625·23-s + 0.769·27-s − 1.11·29-s − 1.61·31-s + 1.39·33-s + 0.640·39-s + 0.780·41-s + 0.914·43-s − 1.31·47-s + 0.840·51-s − 0.824·53-s − 1.04·59-s + 1.02·61-s − 1.71·67-s + 0.722·69-s − 1.30·71-s − 0.234·73-s − 1.01·79-s − 1.22·81-s + 0.658·83-s + 1.28·87-s + 1.16·89-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  =  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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−16.28649092988857, −15.94796458543451, −15.13946540216897, −14.70412494612132, −14.09054929328709, −13.23685422442699, −12.95194231465444, −12.37364634160581, −11.79752379697254, −11.17982207101092, −10.78922758792737, −10.38143566274169, −9.554009025508877, −9.106313701953058, −8.248670694339692, −7.615235371852120, −7.163555273411697, −6.363326339453905, −5.756829109545580, −5.359335761988698, −4.706573506748506, −4.071405169158688, −3.070286787898190, −2.349289777841693, −1.488694742413199, 0, 0, 1.488694742413199, 2.349289777841693, 3.070286787898190, 4.071405169158688, 4.706573506748506, 5.359335761988698, 5.756829109545580, 6.363326339453905, 7.163555273411697, 7.615235371852120, 8.248670694339692, 9.106313701953058, 9.554009025508877, 10.38143566274169, 10.78922758792737, 11.17982207101092, 11.79752379697254, 12.37364634160581, 12.95194231465444, 13.23685422442699, 14.09054929328709, 14.70412494612132, 15.13946540216897, 15.94796458543451, 16.28649092988857

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