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 − 8·31-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s − 45-s − 6·51-s + 6·53-s − 4·57-s − 10·61-s − 2·65-s + 4·67-s − 2·73-s + 75-s + 8·79-s + 81-s + 12·83-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 − 1.43·31-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 0.234·73-s + 0.115·75-s + 0.900·79-s + 1/9·81-s + 1.31·83-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 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.88526111854977, −14.30917243747298, −13.88180867726881, −13.27171311836173, −12.86191520931646, −12.40094956670175, −11.76043143770532, −11.10413990804756, −10.74133386958239, −10.33181916381517, −9.368759884367314, −9.094892801804623, −8.499236818732943, −8.163278278543004, −7.372933174833591, −6.968974413145740, −6.315480206989801, −5.814241196713063, −4.865961383301294, −4.390983419721295, −3.860246728403472, −3.242623757165637, −2.416256054404496, −1.961055625076428, −0.9675309559925506, 0, 0.9675309559925506, 1.961055625076428, 2.416256054404496, 3.242623757165637, 3.860246728403472, 4.390983419721295, 4.865961383301294, 5.814241196713063, 6.315480206989801, 6.968974413145740, 7.372933174833591, 8.163278278543004, 8.499236818732943, 9.094892801804623, 9.368759884367314, 10.33181916381517, 10.74133386958239, 11.10413990804756, 11.76043143770532, 12.40094956670175, 12.86191520931646, 13.27171311836173, 13.88180867726881, 14.30917243747298, 14.88526111854977

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