Properties

Label 2-3528-504.229-c0-0-2
Degree $2$
Conductor $3528$
Sign $0.971 - 0.235i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 + 1.5i)5-s + (−0.866 + 0.5i)6-s − 8-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)10-s + (0.866 − 0.5i)12-s + (1.5 + 0.866i)15-s + 16-s + (−0.499 + 0.866i)18-s + (0.866 − 1.5i)19-s + (0.866 + 1.5i)20-s + (0.5 + 0.866i)23-s + (−0.866 + 0.5i)24-s + ⋯
L(s)  = 1  − 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 + 1.5i)5-s + (−0.866 + 0.5i)6-s − 8-s + (0.499 − 0.866i)9-s + (−0.866 − 1.5i)10-s + (0.866 − 0.5i)12-s + (1.5 + 0.866i)15-s + 16-s + (−0.499 + 0.866i)18-s + (0.866 − 1.5i)19-s + (0.866 + 1.5i)20-s + (0.5 + 0.866i)23-s + (−0.866 + 0.5i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381339472\)
\(L(\frac12)\) \(\approx\) \(1.381339472\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 \)
good5 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.73T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.039928217849024582493904890522, −7.86603733682174083121396316367, −7.38266419910835152361691571060, −6.75603483608755255501974197502, −6.28682503289757458251499592501, −5.26430270649537209195397563684, −3.56784654393501426486002407151, −2.88151665973368974295905410272, −2.34246982427643622087069820684, −1.32034395543754513661483588273, 1.21458692847495199011085181740, 1.94375408824720839235719744808, 2.95883535136583688738117303430, 4.02478695232244343387917314903, 5.02613504011627037139446590308, 5.66814827901407948103492125458, 6.57886958433010366035368021234, 7.75295161268803583067242832865, 8.106274863983535447951220987993, 8.974580754702294557430528350514

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