Properties

Label 2-3528-504.299-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.967 - 0.252i$
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 + (0.991 − 0.130i)6-s + 0.999i·8-s + (0.707 + 0.707i)9-s + 0.517i·11-s + (−0.793 + 0.608i)12-s + (−0.5 − 0.866i)16-s + (−0.991 − 1.71i)17-s + (−0.965 − 0.258i)18-s + (1.37 + 0.793i)19-s + (−0.258 − 0.448i)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 + (0.991 − 0.130i)6-s + 0.999i·8-s + (0.707 + 0.707i)9-s + 0.517i·11-s + (−0.793 + 0.608i)12-s + (−0.5 − 0.866i)16-s + (−0.991 − 1.71i)17-s + (−0.965 − 0.258i)18-s + (1.37 + 0.793i)19-s + (−0.258 − 0.448i)22-s + (0.382 − 0.923i)24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.967 - 0.252i$
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: \((2,\ 3528,\ (\ :0),\ 0.967 - 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5986747933\)
\(L(\frac12)\) \(\approx\) \(0.5986747933\)
\(L(1)\) 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 \( 1 - 0.517iT - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.991 + 1.71i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.37 - 0.793i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (-1.05 + 0.608i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.226 - 0.130i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.785795649745171353844443102708, −7.76609139506327053046847289892, −7.33675520268325511287644316508, −6.66957815824641080134453236311, −6.01949262919036175104024512099, −5.01648908325864214157439009399, −4.73676360745460368889339644443, −3.00505474515213678379614519309, −1.89641692748673062683875493275, −0.836803446379687861155879504916, 0.792985221874233558072367549398, 1.94957590731367717644892062451, 3.22976293152978292107252975663, 3.93324143418424258787826695162, 4.89759524300152514912158234641, 5.80607606307789090096802637337, 6.66804222252789746870418932948, 7.12344412284128797346052381249, 8.221066825501407705395401295342, 8.818852522230828796230487842537

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