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Properties

Label 2-3528-504.403-c0-0-8
Degree $2$
Conductor $3528$
Sign $0.841 + 0.540i$
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 + 0.5i)6-s + 8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.5i)12-s + 16-s + (0.866 − 1.5i)17-s + (0.499 − 0.866i)18-s + (−0.866 − 1.5i)19-s + (−0.5 − 0.866i)22-s + (−0.866 + 0.5i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + 2-s + (−0.866 + 0.5i)3-s + 4-s + (−0.866 + 0.5i)6-s + 8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.866 + 0.5i)12-s + 16-s + (0.866 − 1.5i)17-s + (0.499 − 0.866i)18-s + (−0.866 − 1.5i)19-s + (−0.5 − 0.866i)22-s + (−0.866 + 0.5i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.841 + 0.540i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.841 + 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.796607040\)
\(L(\frac12)\) \(\approx\) \(1.796607040\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.660870704582143721567790469604, −7.70033081168696871301617812530, −6.92672393293560791741765083945, −6.23485492345821431122824332126, −5.57866585753006904270252893606, −4.85409548575324260322186074397, −4.32420235388087467493381312829, −3.22467670364229765008594810217, −2.55336194721316292126627412318, −0.859911790455046369412508870855, 1.58926033695801821960870656194, 2.14525653474081361164154021050, 3.61088685752560499873008458928, 4.21718260032973309155577624806, 5.21975086219487835120749084357, 5.78752159459201593395873178730, 6.30896497352660557979091381012, 7.30273730892601407489007865453, 7.68609452810884148376438754172, 8.521104770443299697256267005205

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