Properties

Degree $2$
Conductor $9680$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s − 4·7-s + 9-s + 4·13-s + 2·15-s − 4·19-s − 8·21-s + 6·23-s + 25-s − 4·27-s + 6·29-s − 8·31-s − 4·35-s + 2·37-s + 8·39-s − 6·41-s + 8·43-s + 45-s − 6·47-s + 9·49-s − 6·53-s − 8·57-s + 12·59-s − 2·61-s − 4·63-s + 4·65-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.10·13-s + 0.516·15-s − 0.917·19-s − 1.74·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.328·37-s + 1.28·39-s − 0.937·41-s + 1.21·43-s + 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.503·63-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9680\)    =    \(2^{4} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{9680} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.822643553\)
\(L(\frac12)\) \(\approx\) \(2.822643553\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.74547014776510, −15.87299457960641, −15.67395117119769, −14.85164570743648, −14.40674426003771, −13.73266617567689, −13.23395971233555, −12.87740616986758, −12.36226217270430, −11.22291469090391, −10.81051766086484, −9.996387910198390, −9.460899590182410, −8.976934047643824, −8.501128001259868, −7.817739662249609, −6.804805668696702, −6.521420213395954, −5.776681301833111, −4.913430593705302, −3.714075182424256, −3.493167026608493, −2.676368743758336, −1.989542998470727, −0.7361169674431854, 0.7361169674431854, 1.989542998470727, 2.676368743758336, 3.493167026608493, 3.714075182424256, 4.913430593705302, 5.776681301833111, 6.521420213395954, 6.804805668696702, 7.817739662249609, 8.501128001259868, 8.976934047643824, 9.460899590182410, 9.996387910198390, 10.81051766086484, 11.22291469090391, 12.36226217270430, 12.87740616986758, 13.23395971233555, 13.73266617567689, 14.40674426003771, 14.85164570743648, 15.67395117119769, 15.87299457960641, 16.74547014776510

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