Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 11 \cdot 19^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s − 2·7-s − 11-s + 2·13-s − 4·17-s − 25-s − 8·29-s + 8·31-s + 4·35-s − 10·37-s + 8·41-s − 2·43-s + 8·47-s − 3·49-s − 2·53-s + 2·55-s + 12·59-s + 10·61-s − 4·65-s − 12·67-s + 8·71-s + 6·73-s + 2·77-s + 2·79-s − 16·83-s + 8·85-s − 14·89-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.755·7-s − 0.301·11-s + 0.554·13-s − 0.970·17-s − 1/5·25-s − 1.48·29-s + 1.43·31-s + 0.676·35-s − 1.64·37-s + 1.24·41-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.269·55-s + 1.56·59-s + 1.28·61-s − 0.496·65-s − 1.46·67-s + 0.949·71-s + 0.702·73-s + 0.227·77-s + 0.225·79-s − 1.75·83-s + 0.867·85-s − 1.48·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 142956 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142956 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(142956\)    =    \(2^{2} \cdot 3^{2} \cdot 11 \cdot 19^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{142956} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 142956,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7036787840$
$L(\frac12)$  $\approx$  $0.7036787840$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;19\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;19\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + T \)
19 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
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

−13.27998517504297, −12.86138835182869, −12.57878874273790, −11.84888084140897, −11.48600058283197, −11.10451176946132, −10.53888373765424, −10.03802380331652, −9.547517827010959, −8.959054631179533, −8.508163893829714, −8.114219100166644, −7.435899042731261, −7.035572495028349, −6.584693094695073, −5.899816206018027, −5.542730508121112, −4.732002830564585, −4.228977671974347, −3.697407272771279, −3.322352965058699, −2.529814254886539, −2.018760649207711, −1.061817681511914, −0.2766228040293899, 0.2766228040293899, 1.061817681511914, 2.018760649207711, 2.529814254886539, 3.322352965058699, 3.697407272771279, 4.228977671974347, 4.732002830564585, 5.542730508121112, 5.899816206018027, 6.584693094695073, 7.035572495028349, 7.435899042731261, 8.114219100166644, 8.508163893829714, 8.959054631179533, 9.547517827010959, 10.03802380331652, 10.53888373765424, 11.10451176946132, 11.48600058283197, 11.84888084140897, 12.57878874273790, 12.86138835182869, 13.27998517504297

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