Properties

Label 2-3528-72.29-c0-0-1
Degree $2$
Conductor $3528$
Sign $-0.342 - 0.939i$
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  + (−0.866 − 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 + 0.866i)4-s + (0.965 + 1.67i)5-s + (0.965 − 0.258i)6-s − 0.999i·8-s − 1.00i·9-s − 1.93i·10-s + (−0.965 − 0.258i)12-s + (1.22 − 0.707i)13-s + (−1.86 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s + 0.517i·19-s + (−0.965 + 1.67i)20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.707 + 0.707i)3-s + (0.499 + 0.866i)4-s + (0.965 + 1.67i)5-s + (0.965 − 0.258i)6-s − 0.999i·8-s − 1.00i·9-s − 1.93i·10-s + (−0.965 − 0.258i)12-s + (1.22 − 0.707i)13-s + (−1.86 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s + 0.517i·19-s + (−0.965 + 1.67i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7434913353\)
\(L(\frac12)\) \(\approx\) \(0.7434913353\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 \)
good5 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 0.517iT - T^{2} \)
23 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
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 \( 1 + T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
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

−9.310418363385589811653700044158, −8.347176600830065063011597708001, −7.48872302380345665634333888719, −6.68349010497452174647849092554, −6.03422985456628983837544928472, −5.59497812122266217983233952652, −3.91780095623006477603332284466, −3.46354238308780149985539804680, −2.54463815070823346425523365640, −1.44989611560682355084572092759, 0.66536986764426518868922145386, 1.61179856406093683147734750945, 2.19504831882071540426650784558, 4.28532710472405255802317161425, 5.03586534291094215728701517591, 5.78686428832881356159532531747, 6.24311846800578080268650521203, 6.89954179704674264017170366720, 8.146410568741262068808657753141, 8.342086029742726045934728175655

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