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Properties

Label 2-3528-504.149-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.458 - 0.888i$
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  − i·2-s + (−0.258 + 0.965i)3-s − 4-s + (0.965 + 1.67i)5-s + (0.965 + 0.258i)6-s + i·8-s + (−0.866 − 0.499i)9-s + (1.67 − 0.965i)10-s + (0.258 − 0.965i)12-s + (1.22 + 0.707i)13-s + (−1.86 + 0.500i)15-s + 16-s + (−0.499 + 0.866i)18-s + (0.448 + 0.258i)19-s + (−0.965 − 1.67i)20-s + ⋯
L(s)  = 1  − i·2-s + (−0.258 + 0.965i)3-s − 4-s + (0.965 + 1.67i)5-s + (0.965 + 0.258i)6-s + i·8-s + (−0.866 − 0.499i)9-s + (1.67 − 0.965i)10-s + (0.258 − 0.965i)12-s + (1.22 + 0.707i)13-s + (−1.86 + 0.500i)15-s + 16-s + (−0.499 + 0.866i)18-s + (0.448 + 0.258i)19-s + (−0.965 − 1.67i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.286612244\)
\(L(\frac12)\) \(\approx\) \(1.286612244\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.258 - 0.965i)T \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
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 \( 1 - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - 0.517iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.165577967771647470969433990551, −8.581818557452514645912630253593, −7.32031426074496253247137809393, −6.36871653621315665942414871855, −5.88010399070666740681404549541, −4.96846847574743718275150662681, −4.01484396126455789669258825486, −3.21449256421386131795569253312, −2.72664569764473842005391873648, −1.54520793365046715443503950040, 0.907058511574132158144474429475, 1.50218452842280610816521720598, 3.08521346697873599964414926152, 4.38455837995271110714839453538, 5.32753022457308140538263666776, 5.54373084809328130077444053959, 6.27848195622828691187989616760, 7.09288210279048995425797996933, 7.978134683860525229908423003857, 8.492859760754661392526445596426

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