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Properties

Label 2-3528-504.61-c0-0-2
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: $\chi_{3528} (2077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.592 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.277465977\)
\(L(\frac12)\) \(\approx\) \(1.277465977\)
\(L(1)\) 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

−9.031754869800310183975994568935, −8.478395042867604950185231624780, −7.75676827640665986439078706217, −6.57148792164608533836102946178, −5.83106154114315708778899614073, −4.74398552167195693497631915712, −4.02103554152313750338680328820, −3.40735923498670101930583246070, −2.10435219177672741173748001823, −1.78750230081466343373303505926, 0.837924400911943452541031380480, 1.87661085909789139587868928446, 2.84672299103079046344177442774, 4.02338629530088577380863675396, 5.31355355617368246822773334818, 5.93429985663100662231995814733, 6.33318401878037039486264745192, 7.26039048847773597874314743291, 7.911272988385730629707697069753, 8.565353648833305814212474641093

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