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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s − 8·23-s + 25-s − 2·29-s + 35-s − 6·37-s + 6·41-s − 4·43-s + 49-s − 10·53-s + 4·55-s − 12·59-s − 14·61-s + 2·65-s − 12·67-s − 8·71-s + 10·73-s + 4·77-s − 16·79-s + 12·83-s − 2·85-s − 10·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s + 0.169·35-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s − 1.37·53-s + 0.539·55-s − 1.56·59-s − 1.79·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + 1.31·83-s − 0.216·85-s − 1.05·89-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  =  \(1\)
Selberg data  =  \((2,\ 20160,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.82777644969353, −15.52497904572033, −14.64239887249415, −14.29264457322113, −13.74050757664380, −13.44509873443981, −12.52545036881055, −12.03414777306236, −11.62189469836404, −10.92823702366567, −10.45958958421482, −9.689974716585640, −9.220677920998896, −8.787780269994214, −7.959682485402171, −7.523656557434341, −6.660711972286748, −6.166388652172635, −5.687418978893860, −4.811517515110441, −4.203940507305483, −3.556039358419806, −2.777809142554225, −1.662799210006684, −1.429566330321504, 0, 1.429566330321504, 1.662799210006684, 2.777809142554225, 3.556039358419806, 4.203940507305483, 4.811517515110441, 5.687418978893860, 6.166388652172635, 6.660711972286748, 7.523656557434341, 7.959682485402171, 8.787780269994214, 9.220677920998896, 9.689974716585640, 10.45958958421482, 10.92823702366567, 11.62189469836404, 12.03414777306236, 12.52545036881055, 13.44509873443981, 13.74050757664380, 14.29264457322113, 14.64239887249415, 15.52497904572033, 15.82777644969353

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