Properties

Label 2-47040-1.1-c1-0-24
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

Downloads

Learn more about

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

\(L(1)\) \(\approx\) \(1.916050378\)
\(L(\frac12)\) \(\approx\) \(1.916050378\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.65387604365781, −14.14023425140060, −13.61673685633762, −12.86504521608144, −12.63708595877652, −12.29610012200487, −11.33316118438534, −10.89611005771600, −10.60473655108564, −9.729584552075993, −9.351748489401567, −8.940881940837945, −8.004581632129311, −7.765344324026310, −7.337328037366579, −6.815435809340372, −5.751063896718677, −5.371762086367650, −4.870097383348957, −4.029094074986213, −3.400898575995148, −2.951557735286377, −2.242746378072034, −1.439312615784487, −0.4747689337625153, 0.4747689337625153, 1.439312615784487, 2.242746378072034, 2.951557735286377, 3.400898575995148, 4.029094074986213, 4.870097383348957, 5.371762086367650, 5.751063896718677, 6.815435809340372, 7.337328037366579, 7.765344324026310, 8.004581632129311, 8.940881940837945, 9.351748489401567, 9.729584552075993, 10.60473655108564, 10.89611005771600, 11.33316118438534, 12.29610012200487, 12.63708595877652, 12.86504521608144, 13.61673685633762, 14.14023425140060, 14.65387604365781

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