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: $\chi_{3528} (2909, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.585877139\)
\(L(\frac12)\) \(\approx\) \(1.585877139\)
\(L(1)\) 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

−8.993731546236692730749790661666, −7.81319575654353502666234654058, −7.39220712600999010014895167302, −6.68390131472402877847928577096, −5.17199674858627497719643232996, −4.71952266582429179545319882217, −3.50268629685441115190876341914, −3.08286166330424703047554455548, −2.24748951670814615707599313911, −1.16845926179826974997399715363, 1.08569701635602419120883950654, 2.73292651074023791419285258641, 3.45514841073596214632302704648, 4.53624243849330685761795112773, 4.96587858191158388101036909657, 5.80247168813192865002459674298, 6.99685972498136755391336850189, 7.47880589691850122185610784616, 8.071135025896711934139792669569, 8.753411096365043911591002547980

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