Properties

Label 2-504-1.1-c5-0-24
Degree $2$
Conductor $504$
Sign $-1$
Analytic cond. $80.8334$
Root an. cond. $8.99074$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 67.2·5-s − 49·7-s + 17.2·11-s + 807.·13-s − 777.·17-s + 262.·19-s + 1.95e3·23-s + 1.39e3·25-s + 1.51e3·29-s + 5.47e3·31-s + 3.29e3·35-s + 6.58e3·37-s − 2.93e3·41-s − 975.·43-s − 2.29e4·47-s + 2.40e3·49-s − 2.64e3·53-s − 1.15e3·55-s − 2.25e4·59-s + 9.53e3·61-s − 5.42e4·65-s − 6.78e4·67-s + 1.01e4·71-s + 6.01e4·73-s − 842.·77-s − 2.77e4·79-s + 1.34e4·83-s + ⋯
L(s)  = 1  − 1.20·5-s − 0.377·7-s + 0.0428·11-s + 1.32·13-s − 0.652·17-s + 0.166·19-s + 0.768·23-s + 0.445·25-s + 0.335·29-s + 1.02·31-s + 0.454·35-s + 0.791·37-s − 0.272·41-s − 0.0804·43-s − 1.51·47-s + 0.142·49-s − 0.129·53-s − 0.0515·55-s − 0.841·59-s + 0.328·61-s − 1.59·65-s − 1.84·67-s + 0.239·71-s + 1.32·73-s − 0.0162·77-s − 0.500·79-s + 0.214·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(80.8334\)
Root analytic conductor: \(8.99074\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 504,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 + 49T \)
good5 \( 1 + 67.2T + 3.12e3T^{2} \)
11 \( 1 - 17.2T + 1.61e5T^{2} \)
13 \( 1 - 807.T + 3.71e5T^{2} \)
17 \( 1 + 777.T + 1.41e6T^{2} \)
19 \( 1 - 262.T + 2.47e6T^{2} \)
23 \( 1 - 1.95e3T + 6.43e6T^{2} \)
29 \( 1 - 1.51e3T + 2.05e7T^{2} \)
31 \( 1 - 5.47e3T + 2.86e7T^{2} \)
37 \( 1 - 6.58e3T + 6.93e7T^{2} \)
41 \( 1 + 2.93e3T + 1.15e8T^{2} \)
43 \( 1 + 975.T + 1.47e8T^{2} \)
47 \( 1 + 2.29e4T + 2.29e8T^{2} \)
53 \( 1 + 2.64e3T + 4.18e8T^{2} \)
59 \( 1 + 2.25e4T + 7.14e8T^{2} \)
61 \( 1 - 9.53e3T + 8.44e8T^{2} \)
67 \( 1 + 6.78e4T + 1.35e9T^{2} \)
71 \( 1 - 1.01e4T + 1.80e9T^{2} \)
73 \( 1 - 6.01e4T + 2.07e9T^{2} \)
79 \( 1 + 2.77e4T + 3.07e9T^{2} \)
83 \( 1 - 1.34e4T + 3.93e9T^{2} \)
89 \( 1 + 1.20e5T + 5.58e9T^{2} \)
97 \( 1 + 2.86e4T + 8.58e9T^{2} \)
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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

−9.629314795905935918413780618966, −8.621178129407207304023885656272, −8.002332842402755477921973223396, −6.92472577299283338810298725808, −6.12087543698603210981862035714, −4.72789175974615907954250042752, −3.83068470321456031051844240210, −2.92364711999085192770274088194, −1.20356547762687194784091125185, 0, 1.20356547762687194784091125185, 2.92364711999085192770274088194, 3.83068470321456031051844240210, 4.72789175974615907954250042752, 6.12087543698603210981862035714, 6.92472577299283338810298725808, 8.002332842402755477921973223396, 8.621178129407207304023885656272, 9.629314795905935918413780618966

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