Properties

Degree $2$
Conductor $20160$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 2·13-s + 6·17-s + 4·19-s + 25-s − 6·29-s − 4·31-s − 35-s − 2·37-s − 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s − 12·59-s − 2·61-s + 2·65-s − 8·67-s + 14·73-s − 16·79-s + 12·83-s − 6·85-s − 6·89-s − 2·91-s − 4·95-s + 14·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.554·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.169·35-s − 0.328·37-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 0.256·61-s + 0.248·65-s − 0.977·67-s + 1.63·73-s − 1.80·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s − 0.209·91-s − 0.410·95-s + 1.42·97-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

Degree: \(2\)
Conductor: \(20160\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{20160} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 20160,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
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 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−15.91226407605405, −15.28403874934255, −14.85927459985325, −14.37613110516447, −13.77151653583357, −13.27786882342499, −12.47410766704653, −11.94724115431297, −11.80635738304079, −10.85380198114865, −10.52124641466091, −9.702089821482670, −9.346175402595431, −8.559240936382010, −7.946209639849854, −7.392991927776872, −7.094962454704953, −6.070109069562114, −5.378259502443448, −5.062736867527928, −4.112682029997934, −3.497973085871147, −2.892683888616175, −1.874901866976816, −1.120718575223088, 0, 1.120718575223088, 1.874901866976816, 2.892683888616175, 3.497973085871147, 4.112682029997934, 5.062736867527928, 5.378259502443448, 6.070109069562114, 7.094962454704953, 7.392991927776872, 7.946209639849854, 8.559240936382010, 9.346175402595431, 9.702089821482670, 10.52124641466091, 10.85380198114865, 11.80635738304079, 11.94724115431297, 12.47410766704653, 13.27786882342499, 13.77151653583357, 14.37613110516447, 14.85927459985325, 15.28403874934255, 15.91226407605405

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