Properties

Label 2-47040-1.1-c1-0-60
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 − 2·17-s + 4·19-s − 8·23-s + 25-s + 27-s + 2·29-s + 4·33-s − 6·37-s − 2·39-s + 6·41-s + 4·43-s + 45-s − 2·51-s + 10·53-s + 4·55-s + 4·57-s + 12·59-s + 14·61-s − 2·65-s + 12·67-s − 8·69-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 − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.986·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 0.280·51-s + 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.248·65-s + 1.46·67-s − 0.963·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 47040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.685236395\)
\(L(\frac12)\) \(\approx\) \(3.685236395\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−14.49515953018779, −14.14539748960918, −13.68047561206072, −13.19207745418526, −12.46289513776627, −12.07890354334443, −11.58100234434254, −11.02289270835950, −10.13012482159108, −9.920316116209855, −9.427176666715008, −8.746722070794667, −8.455395219983939, −7.682154505398448, −7.072758172996997, −6.721287678594444, −5.940026291605426, −5.474034970221939, −4.711168242797152, −3.922284011704527, −3.742061240527588, −2.666213047212437, −2.230457303848493, −1.482028041152214, −0.6698841086584934, 0.6698841086584934, 1.482028041152214, 2.230457303848493, 2.666213047212437, 3.742061240527588, 3.922284011704527, 4.711168242797152, 5.474034970221939, 5.940026291605426, 6.721287678594444, 7.072758172996997, 7.682154505398448, 8.455395219983939, 8.746722070794667, 9.427176666715008, 9.920316116209855, 10.13012482159108, 11.02289270835950, 11.58100234434254, 12.07890354334443, 12.46289513776627, 13.19207745418526, 13.68047561206072, 14.14539748960918, 14.49515953018779

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