Properties

Label 2-47040-1.1-c1-0-67
Degree $2$
Conductor $47040$
Sign $1$
Analytic cond. $375.616$
Root an. cond. $19.3808$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s + 4·11-s + 6·13-s − 15-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 27-s − 10·29-s + 4·31-s − 4·33-s + 6·37-s − 6·39-s − 6·41-s + 4·43-s + 45-s − 12·47-s − 6·51-s − 6·53-s + 4·55-s + 4·57-s − 4·59-s − 2·61-s + 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s − 0.696·33-s + 0.986·37-s − 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s − 0.840·51-s − 0.824·53-s + 0.539·55-s + 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.744·65-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: $1$
Analytic conductor: \(375.616\)
Root analytic conductor: \(19.3808\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47040} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 47040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.058761654\)
\(L(\frac12)\) \(\approx\) \(3.058761654\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
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

−14.64379037632096, −14.07012801637751, −13.51624298299225, −12.92430268869563, −12.72132775572923, −11.91716594203712, −11.42555827642020, −10.97813720464688, −10.65817502260631, −9.771109249408326, −9.453469581624895, −8.906576042011783, −8.282448626061115, −7.739956820021511, −6.905584371387554, −6.479617911021403, −6.037241982381170, −5.506445369136337, −4.853432334854833, −4.129460896559364, −3.524000927079925, −3.050972773050760, −1.818260327497605, −1.361830678912081, −0.7131135993202221, 0.7131135993202221, 1.361830678912081, 1.818260327497605, 3.050972773050760, 3.524000927079925, 4.129460896559364, 4.853432334854833, 5.506445369136337, 6.037241982381170, 6.479617911021403, 6.905584371387554, 7.739956820021511, 8.282448626061115, 8.906576042011783, 9.453469581624895, 9.771109249408326, 10.65817502260631, 10.97813720464688, 11.42555827642020, 11.91716594203712, 12.72132775572923, 12.92430268869563, 13.51624298299225, 14.07012801637751, 14.64379037632096

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