Properties

Degree 2
Conductor $ 2^{6} \cdot 3 \cdot 5^{2} \cdot 11 $
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 + 2·7-s + 9-s − 11-s − 4·13-s + 6·17-s − 4·19-s − 2·21-s + 6·23-s − 27-s − 6·29-s − 8·31-s + 33-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 + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-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 + 0.174·33-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(52800\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{52800} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 52800,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9888569856$
$L(\frac12)$  $\approx$  $0.9888569856$
$L(\frac{3}{2})$   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 \( 1 + T \)
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

−14.48750570849119, −14.16981722730779, −13.21498817800304, −12.92514180328683, −12.35062360516674, −11.97214999297315, −11.32083682423414, −10.89389415331284, −10.46208258093391, −9.843059835897315, −9.359036739878801, −8.728194060745225, −8.071816477444213, −7.560560133310865, −7.122694022112454, −6.590443226616901, −5.617348367798935, −5.392678945483431, −4.875615062893466, −4.256399314858698, −3.444246388819000, −2.878054799049608, −1.842770383814984, −1.522473234852986, −0.3517732063552715, 0.3517732063552715, 1.522473234852986, 1.842770383814984, 2.878054799049608, 3.444246388819000, 4.256399314858698, 4.875615062893466, 5.392678945483431, 5.617348367798935, 6.590443226616901, 7.122694022112454, 7.560560133310865, 8.071816477444213, 8.728194060745225, 9.359036739878801, 9.843059835897315, 10.46208258093391, 10.89389415331284, 11.32083682423414, 11.97214999297315, 12.35062360516674, 12.92514180328683, 13.21498817800304, 14.16981722730779, 14.48750570849119

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