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 − 2·11-s + 2·13-s − 15-s + 4·17-s + 8·23-s + 25-s − 27-s + 2·31-s + 2·33-s − 8·37-s − 2·39-s + 2·41-s + 2·43-s + 45-s − 10·47-s − 4·51-s + 2·53-s − 2·55-s + 4·59-s − 10·61-s + 2·65-s − 2·67-s − 8·69-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.970·17-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.359·31-s + 0.348·33-s − 1.31·37-s − 0.320·39-s + 0.312·41-s + 0.304·43-s + 0.149·45-s − 1.45·47-s − 0.560·51-s + 0.274·53-s − 0.269·55-s + 0.520·59-s − 1.28·61-s + 0.248·65-s − 0.244·67-s − 0.963·69-s − 1.42·71-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.88504445233372, −14.37639370138233, −13.65500778906895, −13.30780713856035, −12.85856903536402, −12.21843508175140, −11.84320792860636, −11.13425930504346, −10.62094643661730, −10.39512031074278, −9.621902108210604, −9.204267435269769, −8.532456848788997, −7.989060123894422, −7.308798839938661, −6.851739626434741, −6.179612399185328, −5.692716067053833, −5.070231703360346, −4.756211229582697, −3.779947560388906, −3.171281318093097, −2.566177832821051, −1.547123377620464, −1.065038958160227, 0, 1.065038958160227, 1.547123377620464, 2.566177832821051, 3.171281318093097, 3.779947560388906, 4.756211229582697, 5.070231703360346, 5.692716067053833, 6.179612399185328, 6.851739626434741, 7.308798839938661, 7.989060123894422, 8.532456848788997, 9.204267435269769, 9.621902108210604, 10.39512031074278, 10.62094643661730, 11.13425930504346, 11.84320792860636, 12.21843508175140, 12.85856903536402, 13.30780713856035, 13.65500778906895, 14.37639370138233, 14.88504445233372

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