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Properties

Label 2-3528-56.51-c0-0-4
Degree $2$
Conductor $3528$
Sign $-0.991 - 0.126i$
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 + (−0.499 − 0.866i)4-s − 0.999·8-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)16-s − 1.99·22-s + (−0.5 − 0.866i)25-s + (0.499 + 0.866i)32-s − 2·43-s + (−0.999 + 1.73i)44-s − 0.999·50-s + 0.999·64-s + (−1 − 1.73i)67-s + (−1 + 1.73i)86-s + (0.999 + 1.73i)88-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + (−1 − 1.73i)11-s + (−0.5 + 0.866i)16-s − 1.99·22-s + (−0.5 − 0.866i)25-s + (0.499 + 0.866i)32-s − 2·43-s + (−0.999 + 1.73i)44-s − 0.999·50-s + 0.999·64-s + (−1 − 1.73i)67-s + (−1 + 1.73i)86-s + (0.999 + 1.73i)88-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9695879801\)
\(L(\frac12)\) \(\approx\) \(0.9695879801\)
\(L(1)\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.445083235320813996020907802080, −7.87742902625199362523657061016, −6.59697450785000371193694496398, −5.91800724205383763113232228712, −5.28638409699505385508433620772, −4.46868294677704997994827425708, −3.43084779623484295203855753939, −2.92177008504583755043945183811, −1.84665010616777858824728853159, −0.45089173415561438820596481864, 1.91780899580110526910302843824, 2.93617752672696660285027792109, 3.92666724090627853094105128638, 4.78820787998930189007439371839, 5.24902643120829221504967623541, 6.15554936294227981925344633043, 7.07125722992744299467600493364, 7.44901414019852637275647194798, 8.177447599615119292780337015819, 8.972404168941035647368013574691

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