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Properties

Label 2-3528-504.299-c0-0-3
Degree $2$
Conductor $3528$
Sign $0.999 + 0.00855i$
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 + (−0.923 + 0.382i)3-s + (0.499 − 0.866i)4-s + (−0.608 + 0.793i)6-s − 0.999i·8-s + (0.707 − 0.707i)9-s + 1.93i·11-s + (−0.130 + 0.991i)12-s + (−0.5 − 0.866i)16-s + (0.608 + 1.05i)17-s + (0.258 − 0.965i)18-s + (−0.226 − 0.130i)19-s + (0.965 + 1.67i)22-s + (0.382 + 0.923i)24-s + 25-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.923 + 0.382i)3-s + (0.499 − 0.866i)4-s + (−0.608 + 0.793i)6-s − 0.999i·8-s + (0.707 − 0.707i)9-s + 1.93i·11-s + (−0.130 + 0.991i)12-s + (−0.5 − 0.866i)16-s + (0.608 + 1.05i)17-s + (0.258 − 0.965i)18-s + (−0.226 − 0.130i)19-s + (0.965 + 1.67i)22-s + (0.382 + 0.923i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.00855i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.999 + 0.00855i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.614036897\)
\(L(\frac12)\) \(\approx\) \(1.614036897\)
\(L(1)\) 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 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 - 1.93iT - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.608 - 1.05i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.226 + 0.130i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.793 - 1.37i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.991 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.923 + 1.60i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.37 + 0.793i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.061350979824586307124480097045, −7.75341212326892945386487083675, −6.93538985520167913746605066334, −6.40639010992344142316917841616, −5.55410576164901694812545268913, −4.80200084240021305645407505183, −4.32800178522911004346513613347, −3.47904135966012681327620473719, −2.23920510060906541986671976859, −1.27565428476547551069688612606, 0.921493865745403836527414212466, 2.49769727647164410105879985559, 3.35895511368304457402626432246, 4.29381574095543254058237399404, 5.28482126709549358436843903890, 5.64831966427323050854033196896, 6.38959119434462281014770008655, 7.05680123394557450449115691713, 7.79864848254603335780348340173, 8.486549748989926426170999804149

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