Properties

Label 2-504-1.1-c5-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 35.5·5-s + 49·7-s − 565.·11-s + 983.·13-s − 200.·17-s + 828.·19-s − 4.43e3·23-s − 1.86e3·25-s + 3.71e3·29-s + 992.·31-s − 1.74e3·35-s − 8.35e3·37-s + 1.34e4·41-s + 298.·43-s + 1.87e4·47-s + 2.40e3·49-s − 1.60e4·53-s + 2.00e4·55-s − 1.27e4·59-s − 3.49e4·61-s − 3.49e4·65-s + 1.19e4·67-s + 1.29e4·71-s + 8.11e4·73-s − 2.77e4·77-s + 4.69e4·79-s + 1.11e5·83-s + ⋯
L(s)  = 1  − 0.635·5-s + 0.377·7-s − 1.40·11-s + 1.61·13-s − 0.167·17-s + 0.526·19-s − 1.74·23-s − 0.596·25-s + 0.820·29-s + 0.185·31-s − 0.240·35-s − 1.00·37-s + 1.25·41-s + 0.0246·43-s + 1.23·47-s + 0.142·49-s − 0.784·53-s + 0.895·55-s − 0.476·59-s − 1.20·61-s − 1.02·65-s + 0.326·67-s + 0.304·71-s + 1.78·73-s − 0.532·77-s + 0.847·79-s + 1.77·83-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: \(0\)
Selberg data: \((2,\ 504,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.612952427\)
\(L(\frac12)\) \(\approx\) \(1.612952427\)
\(L(\frac{7}{2})\) 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 \( 1 + 35.5T + 3.12e3T^{2} \)
11 \( 1 + 565.T + 1.61e5T^{2} \)
13 \( 1 - 983.T + 3.71e5T^{2} \)
17 \( 1 + 200.T + 1.41e6T^{2} \)
19 \( 1 - 828.T + 2.47e6T^{2} \)
23 \( 1 + 4.43e3T + 6.43e6T^{2} \)
29 \( 1 - 3.71e3T + 2.05e7T^{2} \)
31 \( 1 - 992.T + 2.86e7T^{2} \)
37 \( 1 + 8.35e3T + 6.93e7T^{2} \)
41 \( 1 - 1.34e4T + 1.15e8T^{2} \)
43 \( 1 - 298.T + 1.47e8T^{2} \)
47 \( 1 - 1.87e4T + 2.29e8T^{2} \)
53 \( 1 + 1.60e4T + 4.18e8T^{2} \)
59 \( 1 + 1.27e4T + 7.14e8T^{2} \)
61 \( 1 + 3.49e4T + 8.44e8T^{2} \)
67 \( 1 - 1.19e4T + 1.35e9T^{2} \)
71 \( 1 - 1.29e4T + 1.80e9T^{2} \)
73 \( 1 - 8.11e4T + 2.07e9T^{2} \)
79 \( 1 - 4.69e4T + 3.07e9T^{2} \)
83 \( 1 - 1.11e5T + 3.93e9T^{2} \)
89 \( 1 - 3.47e4T + 5.58e9T^{2} \)
97 \( 1 - 9.26e4T + 8.58e9T^{2} \)
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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

−10.36642534538784457901632765727, −9.138289492770311183793425115355, −8.012821283629397858902762720153, −7.83270108465493667520821741688, −6.36233767872178641266434230549, −5.47788046569421298790724544548, −4.32142900050270701478367841349, −3.37962327418072056594856622378, −2.05373339389832616561133157460, −0.63194354444668070666829272954, 0.63194354444668070666829272954, 2.05373339389832616561133157460, 3.37962327418072056594856622378, 4.32142900050270701478367841349, 5.47788046569421298790724544548, 6.36233767872178641266434230549, 7.83270108465493667520821741688, 8.012821283629397858902762720153, 9.138289492770311183793425115355, 10.36642534538784457901632765727

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