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Properties

Label 2-3528-56.5-c0-0-3
Degree $2$
Conductor $3528$
Sign $0.553 + 0.832i$
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 + (0.499 + 0.866i)4-s − 0.999i·8-s + (1.73 − i)11-s + (−0.5 + 0.866i)16-s − 1.99·22-s + (0.5 + 0.866i)25-s − 2i·29-s + (0.866 − 0.499i)32-s + (1.73 + 0.999i)44-s − 0.999i·50-s + (−1.73 + i)53-s + (−1 + 1.73i)58-s − 0.999·64-s + (1 − 1.73i)79-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 0.999i·8-s + (1.73 − i)11-s + (−0.5 + 0.866i)16-s − 1.99·22-s + (0.5 + 0.866i)25-s − 2i·29-s + (0.866 − 0.499i)32-s + (1.73 + 0.999i)44-s − 0.999i·50-s + (−1.73 + i)53-s + (−1 + 1.73i)58-s − 0.999·64-s + (1 − 1.73i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9400372127\)
\(L(\frac12)\) \(\approx\) \(0.9400372127\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.819800751090170398220196387001, −8.041795156796035992766163628034, −7.32982342442018564131054445550, −6.41233943139267317277673759577, −6.00038325677020310668582244367, −4.56196640887765899475342695365, −3.73733318327453000239573023905, −3.06422339134968997396719792631, −1.86896117990721474737616970524, −0.884844544768888568296854984541, 1.20832932415528453541867286532, 2.02889760666804390379638859042, 3.32191514592374821502674757866, 4.43852598387056006457517288443, 5.14179899529117137870446576027, 6.22873198247952456757762136805, 6.77243227741478777210146094625, 7.25850041111212486226869521197, 8.247949018555041251117524935467, 8.872082241169776238577686719079

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