Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 14-s + 16-s + 6·17-s − 8·19-s + 20-s + 25-s − 28-s − 6·29-s + 4·31-s − 32-s − 6·34-s − 35-s + 10·37-s + 8·38-s − 40-s − 6·41-s − 4·43-s + 49-s − 50-s + 6·53-s + 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 1.83·19-s + 0.223·20-s + 1/5·25-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.169·35-s + 1.64·37-s + 1.29·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.141·50-s + 0.824·53-s + 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(106470\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{106470} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 106470,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
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

−13.75864340537045, −13.56068915969030, −12.81463969519722, −12.47386175242674, −12.04406448585310, −11.38923426499475, −10.83048424932188, −10.48079680131269, −9.967882462991365, −9.442464057028176, −9.199524477636392, −8.403888691800619, −8.032943633220067, −7.602769442433091, −6.873290890595641, −6.298465805137839, −6.118422296824051, −5.344602927956544, −4.820342523029125, −3.987867865753555, −3.503619253256403, −2.762567321676444, −2.218046276458991, −1.570272950974728, −0.8352540506298705, 0, 0.8352540506298705, 1.570272950974728, 2.218046276458991, 2.762567321676444, 3.503619253256403, 3.987867865753555, 4.820342523029125, 5.344602927956544, 6.118422296824051, 6.298465805137839, 6.873290890595641, 7.602769442433091, 8.032943633220067, 8.403888691800619, 9.199524477636392, 9.442464057028176, 9.967882462991365, 10.48079680131269, 10.83048424932188, 11.38923426499475, 12.04406448585310, 12.47386175242674, 12.81463969519722, 13.56068915969030, 13.75864340537045

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