↑

Properties

Label 2-539-77.75-c0-0-0
Degree $2$
Conductor $539$
Sign $0.212 + 0.977i$
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  + (−1.08 − 1.20i)2-s + (−0.169 + 1.60i)4-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.169 − 1.60i)18-s − 1.61·22-s + (0.809 − 1.40i)23-s + (0.913 + 0.406i)25-s + (−0.5 − 0.363i)29-s + (−0.499 − 0.866i)32-s + (−1.30 + 0.951i)36-s + (−1.47 + 0.658i)37-s + 0.618·43-s + (1.08 + 1.20i)44-s + ⋯
L(s)  = 1  + (−1.08 − 1.20i)2-s + (−0.169 + 1.60i)4-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (0.669 − 0.743i)11-s + (0.169 − 1.60i)18-s − 1.61·22-s + (0.809 − 1.40i)23-s + (0.913 + 0.406i)25-s + (−0.5 − 0.363i)29-s + (−0.499 − 0.866i)32-s + (−1.30 + 0.951i)36-s + (−1.47 + 0.658i)37-s + 0.618·43-s + (1.08 + 1.20i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(0.268996\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :0),\ 0.212 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5579853527\)
\(L(\frac12)\) \(\approx\) \(0.5579853527\)
\(L(1)\) 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 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (1.08 + 1.20i)T + (-0.104 + 0.994i)T^{2} \)
3 \( 1 + (-0.669 - 0.743i)T^{2} \)
5 \( 1 + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.978 - 0.207i)T^{2} \)
23 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.913 + 0.406i)T^{2} \)
37 \( 1 + (1.47 - 0.658i)T + (0.669 - 0.743i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (0.604 - 0.128i)T + (0.913 - 0.406i)T^{2} \)
59 \( 1 + (0.978 + 0.207i)T^{2} \)
61 \( 1 + (-0.913 - 0.406i)T^{2} \)
67 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.978 + 0.207i)T^{2} \)
79 \( 1 + (-0.413 - 0.459i)T + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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

−10.76663677543434471035684408091, −10.14588307531355794328604802057, −9.112730012091292098084498400683, −8.571168375545114458189216654487, −7.58444946039200205318400713811, −6.50782812891847598903138385791, −5.00637823702467795140908556027, −3.72005970911072326777262703267, −2.57380846926098713644529290350, −1.27703244638978597176428685725, 1.40650063856481072446570543503, 3.58815081625988808005063088001, 4.96121847149860175418137347947, 6.09617435454435077902282568940, 7.03992223580310201867856722267, 7.37522175341555015379022539093, 8.674390525074318109268365292761, 9.292384761094639674019140428923, 9.907110274260967536086847453591, 10.89502097316186048417540828122

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