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Properties

Label 2-3528-56.11-c0-0-2
Degree $2$
Conductor $3528$
Sign $-0.126 - 0.991i$
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 + (0.499 − 0.866i)4-s + 0.999i·8-s + (−0.5 − 0.866i)16-s + (−1.73 + i)23-s + (−0.5 + 0.866i)25-s + 2i·29-s + (0.866 + 0.499i)32-s + 2·43-s + (0.999 − 1.73i)46-s − 0.999i·50-s + (1.73 + i)53-s + (−1 − 1.73i)58-s − 0.999·64-s + (−1 + 1.73i)67-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 0.999i·8-s + (−0.5 − 0.866i)16-s + (−1.73 + i)23-s + (−0.5 + 0.866i)25-s + 2i·29-s + (0.866 + 0.499i)32-s + 2·43-s + (0.999 − 1.73i)46-s − 0.999i·50-s + (1.73 + i)53-s + (−1 − 1.73i)58-s − 0.999·64-s + (−1 + 1.73i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.126 - 0.991i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.126 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6749876219\)
\(L(\frac12)\) \(\approx\) \(0.6749876219\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.041884354360927995675686271386, −8.166729037206226318813117642315, −7.46744842439284807959725137113, −7.00355294551870355843670423358, −5.86450871612867094783185079993, −5.60087180951496049321653079742, −4.45864267817272214961715279780, −3.41647475268109924872651457538, −2.23336398849964261462552981820, −1.28302581909545144614714593559, 0.54979367246043538268561997030, 2.05821612738075287680971891348, 2.60418763308301780815763430985, 3.92603521447643477319240884873, 4.33281826298262415133299023535, 5.82179247410528133879081085550, 6.35693403688169396917143107802, 7.31212807375396822008451886233, 8.038626926811885304720040714421, 8.455829428580855173545978046739

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