Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 2·13-s + 6·17-s + 8·19-s + 25-s + 6·29-s + 4·31-s − 35-s + 10·37-s + 6·41-s − 4·43-s + 49-s − 6·53-s + 12·59-s + 10·61-s − 2·65-s − 4·67-s + 12·71-s − 10·73-s − 8·79-s − 12·83-s + 6·85-s + 6·89-s + 2·91-s + 8·95-s − 10·97-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.554·13-s + 1.45·17-s + 1.83·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.169·35-s + 1.64·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 0.248·65-s − 0.488·67-s + 1.42·71-s − 1.17·73-s − 0.900·79-s − 1.31·83-s + 0.650·85-s + 0.635·89-s + 0.209·91-s + 0.820·95-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(20160\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{20160} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 20160,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.882224132\)
\(L(\frac12)\)  \(\approx\)  \(2.882224132\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
good11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
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

−15.85173529337019, −15.01832607111135, −14.38768530163172, −14.17399259032602, −13.48989058620126, −12.91467861813773, −12.40274990373577, −11.70719193553173, −11.50462585398799, −10.45782800644294, −9.976965012919583, −9.650845023814583, −9.139230417501719, −8.110042392974442, −7.867459243506564, −7.055493494970731, −6.556135973577038, −5.660787483327006, −5.413291181240898, −4.601135633879062, −3.806445063129115, −2.911504340650458, −2.660134336158454, −1.356889746039160, −0.7810726335234028, 0.7810726335234028, 1.356889746039160, 2.660134336158454, 2.911504340650458, 3.806445063129115, 4.601135633879062, 5.413291181240898, 5.660787483327006, 6.556135973577038, 7.055493494970731, 7.867459243506564, 8.110042392974442, 9.139230417501719, 9.650845023814583, 9.976965012919583, 10.45782800644294, 11.50462585398799, 11.70719193553173, 12.40274990373577, 12.91467861813773, 13.48989058620126, 14.17399259032602, 14.38768530163172, 15.01832607111135, 15.85173529337019

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