Properties

Degree $2$
Conductor $16560$
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 + 6·17-s − 4·19-s − 23-s + 25-s + 2·29-s − 2·37-s − 10·41-s + 4·43-s − 7·49-s − 6·53-s − 4·55-s − 4·59-s − 10·61-s + 2·65-s + 12·67-s − 8·71-s + 10·73-s + 8·79-s − 4·83-s − 6·85-s − 18·89-s + 4·95-s + 2·97-s + 101-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.20·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s + 1.46·67-s − 0.949·71-s + 1.17·73-s + 0.900·79-s − 0.439·83-s − 0.650·85-s − 1.90·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16560\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{16560} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 16560,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
23 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.25101789959656, −15.62372895029212, −14.95508211310114, −14.62219732570132, −14.07191010563669, −13.55987255746898, −12.60965838266227, −12.25603796070890, −11.93072077694647, −11.12702454683619, −10.68670282693139, −9.727017200063337, −9.658822227961061, −8.676862119507872, −8.223898777880073, −7.607296730054313, −6.877989165210651, −6.420467761945985, −5.663537571613177, −4.903291206832834, −4.272595860236229, −3.565927997393088, −2.994156637419710, −1.903440192637755, −1.159811446408643, 0, 1.159811446408643, 1.903440192637755, 2.994156637419710, 3.565927997393088, 4.272595860236229, 4.903291206832834, 5.663537571613177, 6.420467761945985, 6.877989165210651, 7.607296730054313, 8.223898777880073, 8.676862119507872, 9.658822227961061, 9.727017200063337, 10.68670282693139, 11.12702454683619, 11.93072077694647, 12.25603796070890, 12.60965838266227, 13.55987255746898, 14.07191010563669, 14.62219732570132, 14.95508211310114, 15.62372895029212, 16.25101789959656

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