Properties

Degree $2$
Conductor $3528$
Sign $0.458 - 0.888i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.965 − 0.258i)3-s − 4-s + (0.258 + 0.448i)5-s + (0.258 − 0.965i)6-s i·8-s + (0.866 + 0.499i)9-s + (−0.448 + 0.258i)10-s + (0.965 + 0.258i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.5i)15-s + 16-s + (−0.499 + 0.866i)18-s + (−1.67 − 0.965i)19-s + (−0.258 − 0.448i)20-s + ⋯
L(s)  = 1  + i·2-s + (−0.965 − 0.258i)3-s − 4-s + (0.258 + 0.448i)5-s + (0.258 − 0.965i)6-s i·8-s + (0.866 + 0.499i)9-s + (−0.448 + 0.258i)10-s + (0.965 + 0.258i)12-s + (1.22 + 0.707i)13-s + (−0.133 − 0.5i)15-s + 16-s + (−0.499 + 0.866i)18-s + (−1.67 − 0.965i)19-s + (−0.258 − 0.448i)20-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8968971641\)
\(L(\frac12)\) \(\approx\) \(0.8968971641\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.93iT - T^{2} \)
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

−8.724735708572200418061843867783, −8.108347220402079394535473680185, −6.95435902575740589738984224205, −6.59757215009832437500825676875, −6.24463864113427233871966541503, −5.18784466894923436666029545332, −4.57619345287390951717060555967, −3.76280385939169173187367722733, −2.31129795111952419094481376649, −0.870459814367495461448571515200, 0.948192522010199699720626754331, 1.78168855134543336327498769590, 3.22372426112599712290023322287, 3.94268517166252701312943840784, 4.75492048615944766054264373507, 5.54140627544451665811059799667, 6.01721830502575943364761008072, 7.06969308487647214701256741454, 8.174719367025333168992135711860, 8.822806961447257352587883587657

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