Properties

Label 2-47040-1.1-c1-0-14
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 − 2·13-s + 15-s + 6·17-s − 8·23-s + 25-s − 27-s − 10·29-s + 8·31-s − 4·33-s − 2·37-s + 2·39-s + 2·41-s − 8·43-s − 45-s − 4·47-s − 6·51-s − 10·53-s − 4·55-s + 4·59-s − 6·61-s + 2·65-s + 8·69-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s − 0.840·51-s − 1.37·53-s − 0.539·55-s + 0.520·59-s − 0.768·61-s + 0.248·65-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: $1$
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

\(L(1)\) \(\approx\) \(1.160958903\)
\(L(\frac12)\) \(\approx\) \(1.160958903\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−14.58970693134674, −14.12347600995005, −13.70043824786718, −12.82898312965420, −12.43212123478318, −11.97579209925128, −11.50580020643662, −11.26497369162240, −10.27363020351574, −9.937048251763307, −9.573527007329480, −8.798373534689449, −8.167300691993155, −7.654024293804415, −7.189017439638925, −6.479924569760775, −5.985716840024006, −5.490596598077741, −4.704460273602861, −4.205280002214714, −3.571101618256422, −3.039737377562841, −1.890063259892580, −1.406474925505371, −0.4083898122833238, 0.4083898122833238, 1.406474925505371, 1.890063259892580, 3.039737377562841, 3.571101618256422, 4.205280002214714, 4.704460273602861, 5.490596598077741, 5.985716840024006, 6.479924569760775, 7.189017439638925, 7.654024293804415, 8.167300691993155, 8.798373534689449, 9.573527007329480, 9.937048251763307, 10.27363020351574, 11.26497369162240, 11.50580020643662, 11.97579209925128, 12.43212123478318, 12.82898312965420, 13.70043824786718, 14.12347600995005, 14.58970693134674

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