Properties

Degree $2$
Conductor $55470$
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-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 3·11-s − 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 3·17-s + 18-s + 19-s + 20-s + 2·21-s − 3·22-s − 6·23-s − 24-s + 25-s − 4·26-s − 27-s − 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.436·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55470\)    =    \(2 \cdot 3 \cdot 5 \cdot 43^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{55470} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 55470,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - T \)
43 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.98332403280032, −14.34960639755194, −13.69920356003369, −13.43009450193388, −12.75323582915808, −12.57758525295383, −11.91690651447910, −11.49432523954256, −10.74751288292874, −10.42459973910764, −9.783742587577754, −9.533515056904200, −8.701603442267652, −7.987406445188273, −7.362573636402387, −6.936366608412977, −6.375690210387134, −5.761391276520825, −5.365741530747513, −4.791767846569997, −4.249531619843229, −3.363289976595874, −2.964662998507557, −2.007170774292416, −1.716687058774999, 0, 0, 1.716687058774999, 2.007170774292416, 2.964662998507557, 3.363289976595874, 4.249531619843229, 4.791767846569997, 5.365741530747513, 5.761391276520825, 6.375690210387134, 6.936366608412977, 7.362573636402387, 7.987406445188273, 8.701603442267652, 9.533515056904200, 9.783742587577754, 10.42459973910764, 10.74751288292874, 11.49432523954256, 11.91690651447910, 12.57758525295383, 12.75323582915808, 13.43009450193388, 13.69920356003369, 14.34960639755194, 14.98332403280032

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