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  =  \((2,\ 20160,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.476189837\)
\(L(\frac12)\)  \(\approx\)  \(2.476189837\)
\(L(\frac{3}{2})\)   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

−15.66372877376916, −14.84227849847947, −14.57222963516721, −14.25906822324724, −13.41232733086373, −12.87826477953670, −12.41155783777465, −11.92281594351689, −11.14104777202373, −10.62831466447460, −10.17124382590511, −9.490692641088322, −9.027433507742835, −8.224299628949460, −7.807787938191166, −7.189064047668526, −6.286390219556861, −6.003912845860439, −5.179695955962125, −4.557992485287182, −3.975707044816319, −2.984069073419609, −2.386892661913681, −1.582958269122756, −0.6597984550590880, 0.6597984550590880, 1.582958269122756, 2.386892661913681, 2.984069073419609, 3.975707044816319, 4.557992485287182, 5.179695955962125, 6.003912845860439, 6.286390219556861, 7.189064047668526, 7.807787938191166, 8.224299628949460, 9.027433507742835, 9.490692641088322, 10.17124382590511, 10.62831466447460, 11.14104777202373, 11.92281594351689, 12.41155783777465, 12.87826477953670, 13.41232733086373, 14.25906822324724, 14.57222963516721, 14.84227849847947, 15.66372877376916

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