↑

Properties

Label 2-539-11.3-c1-0-30
Degree $2$
Conductor $539$
Sign $-0.00480 + 0.999i$
Analytic cond. $4.30393$
Root an. cond. $2.07459$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.78 − 1.29i)2-s + (0.887 − 2.73i)4-s + (−0.594 − 1.83i)8-s + (2.42 − 1.76i)9-s + (−0.0629 − 3.31i)11-s + (1.21 + 0.879i)16-s + (2.04 − 6.29i)18-s + (−4.41 − 5.83i)22-s − 2.56·23-s + (−1.54 − 4.75i)25-s + (−2.42 + 7.45i)29-s + 7.15·32-s + (−2.66 − 8.19i)36-s + (−3.68 + 11.3i)37-s + 12.8·43-s + (−9.11 − 2.77i)44-s + ⋯
L(s)  = 1  + (1.26 − 0.917i)2-s + (0.443 − 1.36i)4-s + (−0.210 − 0.647i)8-s + (0.809 − 0.587i)9-s + (−0.0189 − 0.999i)11-s + (0.302 + 0.219i)16-s + (0.482 − 1.48i)18-s + (−0.941 − 1.24i)22-s − 0.533·23-s + (−0.309 − 0.951i)25-s + (−0.449 + 1.38i)29-s + 1.26·32-s + (−0.443 − 1.36i)36-s + (−0.605 + 1.86i)37-s + 1.95·43-s + (−1.37 − 0.417i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00480 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00480 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.00480 + 0.999i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.00480 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03728 - 2.04710i\)
\(L(\frac12)\) \(\approx\) \(2.03728 - 2.04710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0629 + 3.31i)T \)
good2 \( 1 + (-1.78 + 1.29i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.56T + 23T^{2} \)
29 \( 1 + (2.42 - 7.45i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.68 - 11.3i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.5 - 8.41i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + (-12.9 - 9.43i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.31 - 1.68i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (29.9 - 92.2i)T^{2} \)
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

−10.79805351378078862735254519923, −10.11203180754901750046466664038, −9.012250103734035235780858619680, −7.897981758568970383281901700948, −6.57520150010566266961863667687, −5.75933823655554251664606450202, −4.65185158782486807292084129158, −3.77888840051662535340927308518, −2.85777458604732787151730680954, −1.37261038866111621773013832201, 2.09400770035860424428525695513, 3.76766891314175219725775831652, 4.51256913706758397784003150358, 5.41330005110328416995967819434, 6.34953026518715933240607475674, 7.46024966536423021263012501533, 7.68467165505364032730509158999, 9.307532097516792762994297323396, 10.13450057345268807998480227789, 11.21402788588307997486298875907

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