Properties

Label 2-53040-1.1-c1-0-9
Degree $2$
Conductor $53040$
Sign $1$
Analytic cond. $423.526$
Root an. cond. $20.5797$
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 + 13-s − 15-s + 17-s + 4·19-s + 25-s − 27-s − 2·29-s + 4·33-s + 6·37-s − 39-s − 6·41-s + 4·43-s + 45-s − 7·49-s − 51-s + 6·53-s − 4·55-s − 4·57-s − 4·59-s + 6·61-s + 65-s − 12·67-s + 16·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.696·33-s + 0.986·37-s − 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s − 49-s − 0.140·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.124·65-s − 1.46·67-s + 1.89·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(53040\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(423.526\)
Root analytic conductor: \(20.5797\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 53040,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.683038219\)
\(L(\frac12)\) \(\approx\) \(1.683038219\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
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 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−14.26253144566799, −14.00955253649703, −13.26584602953944, −12.97736127477390, −12.53806558087557, −11.78205123845034, −11.38488051570087, −10.91385561862521, −10.18158354496041, −10.02550692501579, −9.388214716413399, −8.705785076276377, −8.166960425121719, −7.412544286663916, −7.263769946195125, −6.248794451030042, −5.971844822688002, −5.264393312251782, −4.965049155007946, −4.205971526609173, −3.396283577502008, −2.818925514880421, −2.079532454505125, −1.299665685481985, −0.4898067074925211, 0.4898067074925211, 1.299665685481985, 2.079532454505125, 2.818925514880421, 3.396283577502008, 4.205971526609173, 4.965049155007946, 5.264393312251782, 5.971844822688002, 6.248794451030042, 7.263769946195125, 7.412544286663916, 8.166960425121719, 8.705785076276377, 9.388214716413399, 10.02550692501579, 10.18158354496041, 10.91385561862521, 11.38488051570087, 11.78205123845034, 12.53806558087557, 12.97736127477390, 13.26584602953944, 14.00955253649703, 14.26253144566799

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