↑

Properties

Label 2-539-77.18-c0-0-0
Degree $2$
Conductor $539$
Sign $0.458 + 0.888i$
Analytic cond. $0.268996$
Root an. cond. $0.518648$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.89 − 0.198i)2-s + (2.56 + 0.544i)4-s + (−2.92 − 0.951i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (2.95 + 1.31i)16-s + (−0.395 + 1.86i)18-s + 1.90i·22-s + (0.309 + 0.535i)23-s + (−0.669 − 0.743i)25-s + (1.11 − 0.363i)29-s + (−2.66 − 1.53i)32-s + (0.809 − 2.48i)36-s + (0.413 − 0.459i)37-s − 1.17i·43-s + (0.273 − 2.60i)44-s + ⋯
L(s)  = 1  + (−1.89 − 0.198i)2-s + (2.56 + 0.544i)4-s + (−2.92 − 0.951i)8-s + (0.104 − 0.994i)9-s + (−0.104 − 0.994i)11-s + (2.95 + 1.31i)16-s + (−0.395 + 1.86i)18-s + 1.90i·22-s + (0.309 + 0.535i)23-s + (−0.669 − 0.743i)25-s + (1.11 − 0.363i)29-s + (−2.66 − 1.53i)32-s + (0.809 − 2.48i)36-s + (0.413 − 0.459i)37-s − 1.17i·43-s + (0.273 − 2.60i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.458 + 0.888i$
Analytic conductor: \(0.268996\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :0),\ 0.458 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3835623793\)
\(L(\frac12)\) \(\approx\) \(0.3835623793\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 + (0.104 + 0.994i)T \)
good2 \( 1 + (1.89 + 0.198i)T + (0.978 + 0.207i)T^{2} \)
3 \( 1 + (-0.104 + 0.994i)T^{2} \)
5 \( 1 + (0.669 + 0.743i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.978 - 0.207i)T^{2} \)
19 \( 1 + (-0.913 + 0.406i)T^{2} \)
23 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.669 - 0.743i)T^{2} \)
37 \( 1 + (-0.413 + 0.459i)T + (-0.104 - 0.994i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.17iT - T^{2} \)
47 \( 1 + (0.913 - 0.406i)T^{2} \)
53 \( 1 + (-1.47 + 0.658i)T + (0.669 - 0.743i)T^{2} \)
59 \( 1 + (0.913 + 0.406i)T^{2} \)
61 \( 1 + (-0.669 - 0.743i)T^{2} \)
67 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.913 - 0.406i)T^{2} \)
79 \( 1 + (-1.16 - 0.122i)T + (0.978 + 0.207i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.55288567640933022982668685317, −9.968659892168660802743104091760, −9.021539841229717545961410030437, −8.508478017116963373441002781588, −7.55305344329584292386499396788, −6.63416918105864715542169499429, −5.82406688823416810605436963476, −3.69888058083738671154768555179, −2.52301066537474133895342193263, −0.904180718382114109564293680756, 1.61847846942603438357443335395, 2.71942748545024265487422432003, 4.77470590017601454740527618068, 6.09637175358332260048790157044, 7.12776218582333366255824217721, 7.72270928709890730835417269767, 8.535910661761895383937871470784, 9.442410895381852599469720162854, 10.18625507307689840924140278322, 10.77669973774602195352884859780

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