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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s + 9-s + 11-s − 2·13-s − 4·17-s − 6·19-s + 2·21-s − 27-s + 8·29-s + 8·31-s − 33-s + 10·37-s + 2·39-s + 8·41-s + 2·43-s − 8·47-s − 3·49-s + 4·51-s − 2·53-s + 6·57-s + 12·59-s − 10·61-s − 2·63-s − 12·67-s − 8·71-s − 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s − 1.37·19-s + 0.436·21-s − 0.192·27-s + 1.48·29-s + 1.43·31-s − 0.174·33-s + 1.64·37-s + 0.320·39-s + 1.24·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.794·57-s + 1.56·59-s − 1.28·61-s − 0.251·63-s − 1.46·67-s − 0.949·71-s − 0.702·73-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  =  1
Selberg data  =  $(2,\ 52800,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.73483329703672, −14.27392011890522, −13.55114848243676, −13.07969279225607, −12.68640860566816, −12.25238189428295, −11.50599838503049, −11.30302650496279, −10.51894343853786, −10.08369996788506, −9.710599590577355, −8.964979116433684, −8.553083152159401, −7.882658825165032, −7.220204802818011, −6.589098512576162, −6.241543720856790, −5.904977334320913, −4.808907659243286, −4.470594102276826, −4.066004358267467, −2.889670997644496, −2.666669388096461, −1.700057815290368, −0.7810837481897579, 0, 0.7810837481897579, 1.700057815290368, 2.666669388096461, 2.889670997644496, 4.066004358267467, 4.470594102276826, 4.808907659243286, 5.904977334320913, 6.241543720856790, 6.589098512576162, 7.220204802818011, 7.882658825165032, 8.553083152159401, 8.964979116433684, 9.710599590577355, 10.08369996788506, 10.51894343853786, 11.30302650496279, 11.50599838503049, 12.25238189428295, 12.68640860566816, 13.07969279225607, 13.55114848243676, 14.27392011890522, 14.73483329703672

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