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Properties

Label 2-3528-504.317-c0-0-0
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: $\chi_{3528} (2333, \cdot )$
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

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

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