Properties

Label 2-3528-72.29-c0-0-2
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 + (0.258 + 0.448i)5-s + (0.258 + 0.965i)6-s + 0.999i·8-s + 1.00i·9-s + 0.517i·10-s + (−0.258 + 0.965i)12-s + (1.22 − 0.707i)13-s + (−0.133 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s − 1.93i·19-s + (−0.258 + 0.448i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.707 + 0.707i)3-s + (0.499 + 0.866i)4-s + (0.258 + 0.448i)5-s + (0.258 + 0.965i)6-s + 0.999i·8-s + 1.00i·9-s + 0.517i·10-s + (−0.258 + 0.965i)12-s + (1.22 − 0.707i)13-s + (−0.133 + 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.500 + 0.866i)18-s − 1.93i·19-s + (−0.258 + 0.448i)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

\(L(\frac{1}{2})\) \(\approx\) \(2.883203160\)
\(L(\frac12)\) \(\approx\) \(2.883203160\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.93iT - T^{2} \)
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 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
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

−8.843658865022076793192077802637, −8.130315450023434524427952209503, −7.56384219315667568411090331512, −6.59570615369864306793185557844, −5.96548328608091900784780182081, −5.09694961212281206803148217746, −4.37978244184015210288041785365, −3.50285949975553282592395804252, −2.93617984704579203909392690031, −2.00421231405516646786522089724, 1.36241478810149651874768792621, 1.83754609494224675549900643735, 2.98162981875803574789574816959, 3.85917576982379725746295656181, 4.36580316403611913493660313134, 5.79625494132835012097172012986, 6.02307553965640846598594134922, 6.89032356900973158370782244184, 7.80791352098717819694396502803, 8.539287753706819535234253478011

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