Properties

Label 2-504-1.1-c3-0-18
Degree $2$
Conductor $504$
Sign $-1$
Analytic cond. $29.7369$
Root an. cond. $5.45316$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.30·5-s + 7·7-s − 48.9·11-s − 2.60·13-s − 136.·17-s + 45.2·19-s + 38.1·23-s − 85.2·25-s − 52.7·29-s − 14.7·31-s + 44.1·35-s + 333.·37-s − 227.·41-s − 398.·43-s + 184.·47-s + 49·49-s − 359.·53-s − 308.·55-s − 99.9·59-s − 674.·61-s − 16.4·65-s − 376.·67-s + 1.18e3·71-s − 735.·73-s − 342.·77-s − 836.·79-s − 293.·83-s + ⋯
L(s)  = 1  + 0.563·5-s + 0.377·7-s − 1.34·11-s − 0.0556·13-s − 1.95·17-s + 0.545·19-s + 0.345·23-s − 0.682·25-s − 0.337·29-s − 0.0856·31-s + 0.213·35-s + 1.48·37-s − 0.865·41-s − 1.41·43-s + 0.572·47-s + 0.142·49-s − 0.932·53-s − 0.755·55-s − 0.220·59-s − 1.41·61-s − 0.0313·65-s − 0.687·67-s + 1.98·71-s − 1.17·73-s − 0.506·77-s − 1.19·79-s − 0.388·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(29.7369\)
Root analytic conductor: \(5.45316\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 504,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 7T \)
good5 \( 1 - 6.30T + 125T^{2} \)
11 \( 1 + 48.9T + 1.33e3T^{2} \)
13 \( 1 + 2.60T + 2.19e3T^{2} \)
17 \( 1 + 136.T + 4.91e3T^{2} \)
19 \( 1 - 45.2T + 6.85e3T^{2} \)
23 \( 1 - 38.1T + 1.21e4T^{2} \)
29 \( 1 + 52.7T + 2.43e4T^{2} \)
31 \( 1 + 14.7T + 2.97e4T^{2} \)
37 \( 1 - 333.T + 5.06e4T^{2} \)
41 \( 1 + 227.T + 6.89e4T^{2} \)
43 \( 1 + 398.T + 7.95e4T^{2} \)
47 \( 1 - 184.T + 1.03e5T^{2} \)
53 \( 1 + 359.T + 1.48e5T^{2} \)
59 \( 1 + 99.9T + 2.05e5T^{2} \)
61 \( 1 + 674.T + 2.26e5T^{2} \)
67 \( 1 + 376.T + 3.00e5T^{2} \)
71 \( 1 - 1.18e3T + 3.57e5T^{2} \)
73 \( 1 + 735.T + 3.89e5T^{2} \)
79 \( 1 + 836.T + 4.93e5T^{2} \)
83 \( 1 + 293.T + 5.71e5T^{2} \)
89 \( 1 + 1.29e3T + 7.04e5T^{2} \)
97 \( 1 + 201.T + 9.12e5T^{2} \)
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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

−10.09046461708136248517122784480, −9.212162463562009456857222326841, −8.267781750084259463578303960035, −7.37854139940838831305789315317, −6.29890352925165513224438258557, −5.29866504147081077703908520838, −4.43308753737450677353002831174, −2.85074348449569151610321471841, −1.83185373793504048724206216768, 0, 1.83185373793504048724206216768, 2.85074348449569151610321471841, 4.43308753737450677353002831174, 5.29866504147081077703908520838, 6.29890352925165513224438258557, 7.37854139940838831305789315317, 8.267781750084259463578303960035, 9.212162463562009456857222326841, 10.09046461708136248517122784480

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