Properties

Degree $2$
Conductor $3528$
Sign $0.0698 + 0.997i$
Motivic weight $0$
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 + (0.130 + 0.991i)3-s + (0.499 − 0.866i)4-s + (−0.608 − 0.793i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (−1.67 + 0.965i)11-s + (0.923 + 0.382i)12-s + (−0.5 − 0.866i)16-s − 1.21·17-s + (0.707 − 0.707i)18-s − 0.261i·19-s + (0.965 − 1.67i)22-s + (−0.991 + 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.130 + 0.991i)3-s + (0.499 − 0.866i)4-s + (−0.608 − 0.793i)6-s + 0.999i·8-s + (−0.965 + 0.258i)9-s + (−1.67 + 0.965i)11-s + (0.923 + 0.382i)12-s + (−0.5 − 0.866i)16-s − 1.21·17-s + (0.707 − 0.707i)18-s − 0.261i·19-s + (0.965 − 1.67i)22-s + (−0.991 + 0.130i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0698 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.0698 + 0.997i$
Motivic weight: \(0\)
Character: $\chi_{3528} (2939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.0698 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03627808980\)
\(L(\frac12)\) \(\approx\) \(0.03627808980\)
\(L(1)\) 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 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + 1.21T + T^{2} \)
19 \( 1 + 0.261iT - T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
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 \( 1 + T^{2} \)
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 \( 1 + 1.98iT - T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + 1.84T + T^{2} \)
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

−8.609170518831056607338629337566, −7.952350333682190691185708625330, −7.30152099287126395199752979298, −6.39089740773183853470398982125, −5.50704074537300522476729585210, −4.89488356995262423757408514371, −4.15695615852862548057157456897, −2.67744211479729686954756097440, −2.16865519643851876038891156561, −0.02644674351811086044867826727, 1.34336969669496277077143150571, 2.48732685871653939661117690885, 2.90759170237342334303200629344, 4.04506979293518092129440640213, 5.38931882187139872164091866702, 6.10203326944747277448140717851, 6.99498081419462640530809283123, 7.61968539472385521607059972082, 8.251787747429075378547230993550, 8.710670099773310959515922700073

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