Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{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-s − 2·7-s + 9-s − 4·13-s − 6·17-s − 4·19-s − 2·21-s + 6·23-s + 27-s − 6·29-s − 8·31-s + 10·37-s − 4·39-s − 6·41-s − 8·43-s − 6·47-s − 3·49-s − 6·51-s − 4·57-s − 8·61-s − 2·63-s − 4·67-s + 6·69-s − 6·71-s + 2·73-s + 14·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.10·13-s − 1.45·17-s − 0.917·19-s − 0.436·21-s + 1.25·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.840·51-s − 0.529·57-s − 1.02·61-s − 0.251·63-s − 0.488·67-s + 0.722·69-s − 0.712·71-s + 0.234·73-s + 1.57·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(145200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 11^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{145200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(2,\ 145200,\ (\ :1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.64042953036326, −13.19842118219057, −13.11285603659735, −12.56812974635377, −12.06321951717657, −11.33415526528580, −10.99167526134354, −10.53764878200819, −9.838373195946363, −9.415580407630454, −9.161687982994746, −8.612033165892677, −8.020818191562497, −7.480815135312983, −6.966060670830373, −6.578839685571734, −6.117589069032709, −5.248376631312455, −4.838707923784468, −4.289912486916443, −3.641504977004457, −3.140906152489547, −2.490282974240130, −2.059875870830349, −1.354687133117632, 0, 0, 1.354687133117632, 2.059875870830349, 2.490282974240130, 3.140906152489547, 3.641504977004457, 4.289912486916443, 4.838707923784468, 5.248376631312455, 6.117589069032709, 6.578839685571734, 6.966060670830373, 7.480815135312983, 8.020818191562497, 8.612033165892677, 9.161687982994746, 9.415580407630454, 9.838373195946363, 10.53764878200819, 10.99167526134354, 11.33415526528580, 12.06321951717657, 12.56812974635377, 13.11285603659735, 13.19842118219057, 13.64042953036326

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