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Properties

Label 2-504-168.125-c1-0-20
Degree $2$
Conductor $504$
Sign $0.766 - 0.642i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.909 + 1.08i)2-s + (−0.347 + 1.96i)4-s − 3.85i·5-s + (1.87 + 1.86i)7-s + (−2.44 + 1.41i)8-s + (4.17 − 3.49i)10-s + 4.26·11-s − 1.89·13-s + (−0.309 + 3.72i)14-s + (−3.75 − 1.36i)16-s + 6.66·17-s + 2.89·19-s + (7.58 + 1.33i)20-s + (3.87 + 4.62i)22-s + 1.73i·23-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s − 1.72i·5-s + (0.710 + 0.703i)7-s + (−0.866 + 0.500i)8-s + (1.31 − 1.10i)10-s + 1.28·11-s − 0.524·13-s + (−0.0825 + 0.996i)14-s + (−0.939 − 0.342i)16-s + 1.61·17-s + 0.664·19-s + (1.69 + 0.298i)20-s + (0.827 + 0.985i)22-s + 0.361i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03247 + 0.739723i\)
\(L(\frac12)\) \(\approx\) \(2.03247 + 0.739723i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 - 1.08i)T \)
3 \( 1 \)
7 \( 1 + (-1.87 - 1.86i)T \)
good5 \( 1 + 3.85iT - 5T^{2} \)
11 \( 1 - 4.26T + 11T^{2} \)
13 \( 1 + 1.89T + 13T^{2} \)
17 \( 1 - 6.66T + 17T^{2} \)
19 \( 1 - 2.89T + 19T^{2} \)
23 \( 1 - 1.73iT - 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 - 1.29iT - 31T^{2} \)
37 \( 1 + 2.26iT - 37T^{2} \)
41 \( 1 + 8.49T + 41T^{2} \)
43 \( 1 + 2.09iT - 43T^{2} \)
47 \( 1 + 9.89T + 47T^{2} \)
53 \( 1 + 5.05T + 53T^{2} \)
59 \( 1 + 2.67iT - 59T^{2} \)
61 \( 1 + 13.1T + 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 + 15.5iT - 71T^{2} \)
73 \( 1 - 9.43iT - 73T^{2} \)
79 \( 1 - 4.08T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 1.98iT - 97T^{2} \)
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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

−11.65292021860422246686680529646, −9.704295854552197260561511559467, −9.008310674320288475331452277851, −8.277403594840583560767269335088, −7.52520979646241330958783988651, −6.12652344254398793756646338999, −5.21683179770736721489020586462, −4.71753838713565159030158401839, −3.46376645799265727723385368694, −1.47528344160156242900204764143, 1.50111654297309584559273188227, 3.00938223878813106385106037740, 3.72836801384630418232075677267, 4.94346565826326192089602787073, 6.23677749964757814647418801296, 6.95728230036730234682504309487, 7.937326401335013183895958382236, 9.574490232455961500022148261511, 10.16000450484011175820971002925, 10.89966647846332394227494660564

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