Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 $
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 − 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  =  1
Selberg data  =  $(2,\ 2880,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]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

−18.84772944545514, −18.37268744189972, −17.86107860763907, −17.15850621089077, −16.46769410600310, −15.92534365147358, −15.13415596850628, −14.73116943002231, −13.74672091077615, −13.17739019873558, −12.88070780980752, −11.90376046835221, −11.03950107500412, −10.65630686000040, −9.828402187127102, −9.201306093698397, −8.226082691547122, −7.895948826452069, −6.679428376878416, −6.226556658534405, −5.237115829018037, −4.616377860412270, −3.493839422897356, −2.561153813343379, −1.630033043155359, 0, 1.630033043155359, 2.561153813343379, 3.493839422897356, 4.616377860412270, 5.237115829018037, 6.226556658534405, 6.679428376878416, 7.895948826452069, 8.226082691547122, 9.201306093698397, 9.828402187127102, 10.65630686000040, 11.03950107500412, 11.90376046835221, 12.88070780980752, 13.17739019873558, 13.74672091077615, 14.73116943002231, 15.13415596850628, 15.92534365147358, 16.46769410600310, 17.15850621089077, 17.86107860763907, 18.37268744189972, 18.84772944545514

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