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Properties

Label 2-3528-504.331-c0-0-7
Degree $2$
Conductor $3528$
Sign $0.678 + 0.734i$
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  + (−0.5 − 0.866i)2-s + i·3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)6-s + 0.999·8-s − 9-s + 11-s + (−0.866 − 0.499i)12-s + (−0.5 − 0.866i)16-s + (−0.866 − 1.5i)17-s + (0.5 + 0.866i)18-s + (0.866 − 1.5i)19-s + (−0.5 − 0.866i)22-s + 0.999i·24-s + 25-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + i·3-s + (−0.499 + 0.866i)4-s + (0.866 − 0.5i)6-s + 0.999·8-s − 9-s + 11-s + (−0.866 − 0.499i)12-s + (−0.5 − 0.866i)16-s + (−0.866 − 1.5i)17-s + (0.5 + 0.866i)18-s + (0.866 − 1.5i)19-s + (−0.5 − 0.866i)22-s + 0.999i·24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.678 + 0.734i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1843, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.678 + 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9020106957\)
\(L(\frac12)\) \(\approx\) \(0.9020106957\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 - iT \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
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 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.972550626167594234186939324881, −8.331307918414042740266182103164, −7.16621093452617221739547174683, −6.65703327617474012913310203958, −5.12304320534923562618298278107, −4.81662428547783065456760174437, −3.84180064589181188000014588392, −3.08916462947807128443092321381, −2.30921815260248950664228548187, −0.73147836974766753300425314296, 1.21149292652454301434211549745, 1.88463744275839813241971611435, 3.40396233842231186169105520037, 4.35952299403539024848344469871, 5.46665749277556177612323546130, 6.10871872416341367082888780216, 6.68957645211901499637534085912, 7.27165743128968067827041975458, 8.250590609868790518838887343899, 8.473055568583809591988491708163

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