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Properties

Label 2-3528-504.157-c0-0-1
Degree $2$
Conductor $3528$
Sign $0.592 + 0.805i$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + 0.999·12-s + (−1 + 1.73i)13-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (0.5 + 0.866i)19-s + (0.499 + 0.866i)20-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.499 − 0.866i)6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + 0.999·12-s + (−1 + 1.73i)13-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (0.5 + 0.866i)19-s + (0.499 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02834145901\)
\(L(\frac12)\) \(\approx\) \(0.02834145901\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.605852893270088321251759999521, −7.88107479113781708540118643740, −7.19759269325710739546937599720, −6.44611536824688367775281278041, −5.71654111896407470516932760850, −4.74157142594756227394563381850, −4.31062343527203027913334724230, −3.50890274371951980866263204186, −1.83147568710794522010079156713, −0.02515613220077678073874183418, 1.03978766618099246977404762834, 2.42539684432091705781786939488, 3.04963581831301415464755874808, 4.12842748159661787608799036502, 5.01650576265853764239350981387, 5.77551784126851747780892134716, 7.05192716179111697690513014297, 7.56205853281525838035978625803, 8.018444713588179514722949124677, 8.678823072014999364670049501212

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