Properties

Degree 2
Conductor $ 2^{4} \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 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5040\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{5040} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 5040,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
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

−18.12036996008194, −17.18145637280857, −16.91239054886632, −16.22606097644418, −15.58283102336401, −15.07107244767513, −14.43435090690432, −13.79052721046154, −13.07899512074289, −12.39795512712985, −12.10112566061446, −11.08783471697299, −10.65968389731635, −9.999214284873384, −9.159356573107523, −8.583930059748585, −7.859973114851043, −7.249009702472835, −6.350563521343611, −5.838484408318170, −4.902891267334214, −3.990409352946808, −3.453754073466757, −2.439251799703037, −1.310709945745928, 0, 1.310709945745928, 2.439251799703037, 3.453754073466757, 3.990409352946808, 4.902891267334214, 5.838484408318170, 6.350563521343611, 7.249009702472835, 7.859973114851043, 8.583930059748585, 9.159356573107523, 9.999214284873384, 10.65968389731635, 11.08783471697299, 12.10112566061446, 12.39795512712985, 13.07899512074289, 13.79052721046154, 14.43435090690432, 15.07107244767513, 15.58283102336401, 16.22606097644418, 16.91239054886632, 17.18145637280857, 18.12036996008194

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