Properties

Degree $2$
Conductor $45486$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 4·11-s − 6·13-s − 14-s + 16-s − 2·17-s + 2·20-s + 4·22-s − 8·23-s − 25-s − 6·26-s − 28-s − 2·29-s + 32-s − 2·34-s − 2·35-s + 10·37-s + 2·40-s − 6·41-s − 4·43-s + 4·44-s − 8·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 1.20·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.447·20-s + 0.852·22-s − 1.66·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s − 0.371·29-s + 0.176·32-s − 0.342·34-s − 0.338·35-s + 1.64·37-s + 0.316·40-s − 0.937·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45486 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45486 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45486\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{45486} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 45486,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.682674090\)
\(L(\frac12)\) \(\approx\) \(3.682674090\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
19 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.46693955898140, −14.18319892760150, −13.62416033898230, −13.15853571183154, −12.63390415410471, −11.98493677118942, −11.79375579656525, −11.16699364739058, −10.30449464140171, −9.881943953560530, −9.593865608947072, −9.018304895975381, −8.192292784700145, −7.625851762095147, −6.912227842020914, −6.515881105093448, −5.996141961480742, −5.435209209468500, −4.814742947537908, −4.127977881209739, −3.704879412469944, −2.764228955081988, −2.155840740110551, −1.762967993837964, −0.5717430969273195, 0.5717430969273195, 1.762967993837964, 2.155840740110551, 2.764228955081988, 3.704879412469944, 4.127977881209739, 4.814742947537908, 5.435209209468500, 5.996141961480742, 6.515881105093448, 6.912227842020914, 7.625851762095147, 8.192292784700145, 9.018304895975381, 9.593865608947072, 9.881943953560530, 10.30449464140171, 11.16699364739058, 11.79375579656525, 11.98493677118942, 12.63390415410471, 13.15853571183154, 13.62416033898230, 14.18319892760150, 14.46693955898140

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