Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13 $
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 − 5-s − 4·7-s − 8-s + 10-s + 11-s + 13-s + 4·14-s + 16-s + 6·17-s − 4·19-s − 20-s − 22-s + 25-s − 26-s − 4·28-s + 6·29-s − 4·31-s − 32-s − 6·34-s + 4·35-s + 2·37-s + 4·38-s + 40-s + 6·41-s − 4·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.676·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(12870\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{12870} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 12870,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8413358796$
$L(\frac12)$  $\approx$  $0.8413358796$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−16.48246828360342, −15.77628346138738, −15.38159726240380, −14.67170437231359, −14.12068329414112, −13.32208920759921, −12.75440349831566, −12.20269160060603, −11.87553742122930, −10.86633985539304, −10.53100449683640, −9.823972699527472, −9.391632071903364, −8.767304386589642, −8.119883517215831, −7.467742237246035, −6.868940278239574, −6.184477063052261, −5.825873086894794, −4.704008329366774, −3.819542477159719, −3.249799995592812, −2.610964452875049, −1.421285817659045, −0.4806467155302907, 0.4806467155302907, 1.421285817659045, 2.610964452875049, 3.249799995592812, 3.819542477159719, 4.704008329366774, 5.825873086894794, 6.184477063052261, 6.868940278239574, 7.467742237246035, 8.119883517215831, 8.767304386589642, 9.391632071903364, 9.823972699527472, 10.53100449683640, 10.86633985539304, 11.87553742122930, 12.20269160060603, 12.75440349831566, 13.32208920759921, 14.12068329414112, 14.67170437231359, 15.38159726240380, 15.77628346138738, 16.48246828360342

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