Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 $
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 + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 25-s − 2·29-s + 10·37-s − 10·41-s + 4·43-s + 8·47-s − 7·49-s − 10·53-s + 4·55-s + 4·59-s + 2·61-s + 2·65-s + 12·67-s − 8·71-s + 10·73-s − 12·83-s − 2·85-s + 6·89-s + 4·95-s + 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s + 1.64·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s − 49-s − 1.37·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 1.31·83-s − 0.216·85-s + 0.635·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2880} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2880,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.287175190$
$L(\frac12)$  $\approx$  $2.287175190$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−18.54213319135186, −18.19500883663584, −17.24939558219294, −17.05264513883004, −16.13342205931270, −15.65378008338639, −14.75168744030397, −14.26905838379084, −13.61453313321894, −13.01400232142143, −12.24838283790474, −11.43961039065414, −11.09490662587941, −10.03507549618204, −9.475865102875268, −8.881388845996160, −8.084180434635331, −7.168422588620047, −6.444916367191891, −5.838179140732038, −4.881202684168501, −3.994232978203093, −3.167995710440271, −1.983642417170570, −0.9916288656424814, 0.9916288656424814, 1.983642417170570, 3.167995710440271, 3.994232978203093, 4.881202684168501, 5.838179140732038, 6.444916367191891, 7.168422588620047, 8.084180434635331, 8.881388845996160, 9.475865102875268, 10.03507549618204, 11.09490662587941, 11.43961039065414, 12.24838283790474, 13.01400232142143, 13.61453313321894, 14.26905838379084, 14.75168744030397, 15.65378008338639, 16.13342205931270, 17.05264513883004, 17.24939558219294, 18.19500883663584, 18.54213319135186

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