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Properties

Label 2-3528-504.149-c0-0-2
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 + (0.965 + 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: $0.458 - 0.888i$
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

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

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

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