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 + 6·13-s − 2·17-s − 8·19-s + 8·23-s + 25-s − 2·29-s − 4·31-s − 35-s + 2·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s + 10·53-s − 4·59-s + 2·61-s + 6·65-s + 4·67-s − 12·71-s − 2·73-s − 8·79-s + 4·83-s − 2·85-s + 6·89-s − 6·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.66·13-s − 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s − 0.520·59-s + 0.256·61-s + 0.744·65-s + 0.488·67-s − 1.42·71-s − 0.234·73-s − 0.900·79-s + 0.439·83-s − 0.216·85-s + 0.635·89-s − 0.628·91-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  =  $(2,\ 20160,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.393744848$
$L(\frac12)$  $\approx$  $2.393744848$
$L(\frac{3}{2})$   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 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−15.57678487169175, −15.13522704984469, −14.64903602111867, −13.92673635353632, −13.37024571490891, −12.86806787934590, −12.74162580859021, −11.71415470487318, −11.02630184924767, −10.76519329389449, −10.28718143404568, −9.267601898669318, −8.948992614974241, −8.585584672639214, −7.723518881056564, −6.942974054356073, −6.502571673048224, −5.863828128449019, −5.415885834161694, −4.294595334923704, −4.030161285366436, −3.070503431014483, −2.391166227931310, −1.541426529768578, −0.6562282347406039, 0.6562282347406039, 1.541426529768578, 2.391166227931310, 3.070503431014483, 4.030161285366436, 4.294595334923704, 5.415885834161694, 5.863828128449019, 6.502571673048224, 6.942974054356073, 7.723518881056564, 8.585584672639214, 8.948992614974241, 9.267601898669318, 10.28718143404568, 10.76519329389449, 11.02630184924767, 11.71415470487318, 12.74162580859021, 12.86806787934590, 13.37024571490891, 13.92673635353632, 14.64903602111867, 15.13522704984469, 15.57678487169175

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