Properties

Label 2-3528-504.221-c0-0-1
Degree $2$
Conductor $3528$
Sign $0.220 - 0.975i$
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 + (0.258 + 0.965i)3-s + (0.499 + 0.866i)4-s − 0.517·5-s + (0.258 − 0.965i)6-s − 0.999i·8-s + (−0.866 + 0.499i)9-s + (0.448 + 0.258i)10-s + (−0.707 + 0.707i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 18-s + (1.67 − 0.965i)19-s + (−0.258 − 0.448i)20-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.258 + 0.965i)3-s + (0.499 + 0.866i)4-s − 0.517·5-s + (0.258 − 0.965i)6-s − 0.999i·8-s + (−0.866 + 0.499i)9-s + (0.448 + 0.258i)10-s + (−0.707 + 0.707i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.499i)15-s + (−0.5 + 0.866i)16-s + 18-s + (1.67 − 0.965i)19-s + (−0.258 − 0.448i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8214186410\)
\(L(\frac12)\) \(\approx\) \(0.8214186410\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 \)
good5 \( 1 + 0.517T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
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 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
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

−9.082920873392461098713241720575, −8.352031788287855067036978996476, −7.67700868714691183083411916847, −6.98486926723948949878900151316, −5.88941615566807126915712466165, −4.98723108590272737712486745233, −3.79112932349929607899015162466, −3.60706404624817115904923309150, −2.54421220261743916796155680247, −1.26687336731633000569375645655, 0.71169294104140318309667719747, 1.67719399418523739410107668390, 2.84745919807383392357014496457, 3.71789778363650530790848110228, 5.10920250084778789919002868515, 6.00853929654260585757825571137, 6.37666567790245054109427685238, 7.49092553175390996201453501290, 7.72589702376111593609034524114, 8.496948870053982102717133709250

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