Properties

Degree $2$
Conductor $47040$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s + 2·13-s + 15-s + 6·17-s + 4·19-s + 25-s − 27-s + 6·29-s − 4·31-s − 2·37-s − 2·39-s − 6·41-s + 8·43-s − 45-s − 12·47-s − 6·51-s − 6·53-s − 4·57-s + 12·59-s + 2·61-s − 2·65-s + 8·67-s − 14·73-s − 75-s + 16·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 1.21·43-s − 0.149·45-s − 1.75·47-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.248·65-s + 0.977·67-s − 1.63·73-s − 0.115·75-s + 1.80·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47040\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{47040} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47040,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−14.86506402508324, −14.24565395719286, −13.97123538343722, −13.19513347122750, −12.72482810707167, −12.18783347308888, −11.78194918101612, −11.28715781197071, −10.80805384581325, −10.10420976372878, −9.812125490468749, −9.154188056910348, −8.348162492232337, −8.059801703756285, −7.367349231890990, −6.861646288896111, −6.293155984125068, −5.515092132604514, −5.265516304841178, −4.525255801390794, −3.752822661661711, −3.340585967472077, −2.601806589371526, −1.461296582002938, −1.028871347506232, 0, 1.028871347506232, 1.461296582002938, 2.601806589371526, 3.340585967472077, 3.752822661661711, 4.525255801390794, 5.265516304841178, 5.515092132604514, 6.293155984125068, 6.861646288896111, 7.367349231890990, 8.059801703756285, 8.348162492232337, 9.154188056910348, 9.812125490468749, 10.10420976372878, 10.80805384581325, 11.28715781197071, 11.78194918101612, 12.18783347308888, 12.72482810707167, 13.19513347122750, 13.97123538343722, 14.24565395719286, 14.86506402508324

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