Properties

Degree $2$
Conductor $3528$
Sign $0.970 - 0.242i$
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 + (0.793 − 0.608i)3-s + (0.499 + 0.866i)4-s + (0.991 − 0.130i)6-s + 0.999i·8-s + (0.258 − 0.965i)9-s + (−0.448 − 0.258i)11-s + (0.923 + 0.382i)12-s + (−0.5 + 0.866i)16-s + 1.98·17-s + (0.707 − 0.707i)18-s − 1.58i·19-s + (−0.258 − 0.448i)22-s + (0.608 + 0.793i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.793 − 0.608i)3-s + (0.499 + 0.866i)4-s + (0.991 − 0.130i)6-s + 0.999i·8-s + (0.258 − 0.965i)9-s + (−0.448 − 0.258i)11-s + (0.923 + 0.382i)12-s + (−0.5 + 0.866i)16-s + 1.98·17-s + (0.707 − 0.707i)18-s − 1.58i·19-s + (−0.258 − 0.448i)22-s + (0.608 + 0.793i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.970 - 0.242i$
Motivic weight: \(0\)
Character: $\chi_{3528} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.833376249\)
\(L(\frac12)\) \(\approx\) \(2.833376249\)
\(L(1)\) 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 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - 1.98T + T^{2} \)
19 \( 1 + 1.58iT - T^{2} \)
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 \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.608 + 1.05i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.21iT - T^{2} \)
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 \( 1 + (-0.226 - 0.130i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.399544592739904302807555286369, −7.944448097543634624504937744047, −7.26794657517654587814965611468, −6.66712061338354724195162180584, −5.74665398215646335989429598426, −5.11112474747394538850362220865, −4.06226401448649985626430482396, −3.15986994181416274705576819987, −2.71620069194163141556831464289, −1.40881303361777353639516534741, 1.52775688435436435369141385290, 2.42620307081950177388392138754, 3.43857435055484385113814972945, 3.80310658545165585453114999895, 4.84054348237634495395962879527, 5.48770321937187280507957615810, 6.19381480848950290652857662123, 7.45110333569572128547824116047, 7.86412928756774927252737743857, 8.787953042268719559540231666477

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