Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 6·13-s + 2·17-s − 8·19-s + 8·23-s − 27-s − 2·29-s + 4·31-s + 2·37-s + 6·39-s + 6·41-s + 4·43-s − 8·47-s − 2·51-s − 10·53-s + 8·57-s + 4·59-s + 2·61-s + 4·67-s − 8·69-s + 12·71-s − 2·73-s − 8·79-s + 81-s + 4·83-s + 2·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.66·13-s + 0.485·17-s − 1.83·19-s + 1.66·23-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.328·37-s + 0.960·39-s + 0.937·41-s + 0.609·43-s − 1.16·47-s − 0.280·51-s − 1.37·53-s + 1.05·57-s + 0.520·59-s + 0.256·61-s + 0.488·67-s − 0.963·69-s + 1.42·71-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.439·83-s + 0.214·87-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(58800\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{58800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 58800,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.61593535222233, −14.33733035521996, −13.34744145626515, −13.01637281425753, −12.43770024503799, −12.26508813512266, −11.48199529181625, −10.95226620602164, −10.68574674374604, −9.920781100534557, −9.546943640340210, −9.067472034772326, −8.202982953387435, −7.905252410776016, −7.086137409560652, −6.784333575762370, −6.202173972493720, −5.476238013195452, −4.971093496378076, −4.505401806493081, −3.922203121718221, −2.949427931679554, −2.493268692406329, −1.717249155018236, −0.7960587340447650, 0, 0.7960587340447650, 1.717249155018236, 2.493268692406329, 2.949427931679554, 3.922203121718221, 4.505401806493081, 4.971093496378076, 5.476238013195452, 6.202173972493720, 6.784333575762370, 7.086137409560652, 7.905252410776016, 8.202982953387435, 9.067472034772326, 9.546943640340210, 9.920781100534557, 10.68574674374604, 10.95226620602164, 11.48199529181625, 12.26508813512266, 12.43770024503799, 13.01637281425753, 13.34744145626515, 14.33733035521996, 14.61593535222233

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