Properties

Label 2-539-77.16-c1-0-25
Degree $2$
Conductor $539$
Sign $0.0939 + 0.995i$
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  + (0.760 + 0.161i)2-s + (0.0646 − 0.614i)3-s + (−1.27 − 0.567i)4-s + (0.148 + 0.165i)5-s + (0.148 − 0.456i)6-s + (−2.13 − 1.55i)8-s + (2.56 + 0.544i)9-s + (0.0865 + 0.149i)10-s + (2.48 − 2.19i)11-s + (−0.431 + 0.746i)12-s + (−2.01 − 6.20i)13-s + (0.111 − 0.0808i)15-s + (0.494 + 0.548i)16-s + (−4.23 + 0.901i)17-s + (1.85 + 0.827i)18-s + (−2.66 + 1.18i)19-s + ⋯
L(s)  = 1  + (0.537 + 0.114i)2-s + (0.0372 − 0.354i)3-s + (−0.637 − 0.283i)4-s + (0.0665 + 0.0739i)5-s + (0.0606 − 0.186i)6-s + (−0.755 − 0.548i)8-s + (0.853 + 0.181i)9-s + (0.0273 + 0.0473i)10-s + (0.749 − 0.662i)11-s + (−0.124 + 0.215i)12-s + (−0.559 − 1.72i)13-s + (0.0287 − 0.0208i)15-s + (0.123 + 0.137i)16-s + (−1.02 + 0.218i)17-s + (0.438 + 0.195i)18-s + (−0.610 + 0.272i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0939 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0939 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.0939 + 0.995i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.0939 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11655 - 1.01615i\)
\(L(\frac12)\) \(\approx\) \(1.11655 - 1.01615i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.48 + 2.19i)T \)
good2 \( 1 + (-0.760 - 0.161i)T + (1.82 + 0.813i)T^{2} \)
3 \( 1 + (-0.0646 + 0.614i)T + (-2.93 - 0.623i)T^{2} \)
5 \( 1 + (-0.148 - 0.165i)T + (-0.522 + 4.97i)T^{2} \)
13 \( 1 + (2.01 + 6.20i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.23 - 0.901i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (2.66 - 1.18i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (-1.94 + 3.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.05 + 2.21i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.61 + 5.11i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-0.590 - 5.62i)T + (-36.1 + 7.69i)T^{2} \)
41 \( 1 + (-1.08 - 0.786i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + (5.52 - 2.45i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (1.14 - 1.27i)T + (-5.54 - 52.7i)T^{2} \)
59 \( 1 + (-8.70 - 3.87i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-6.44 - 7.15i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (0.635 + 1.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.87 - 8.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.10 + 2.27i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (-4.43 - 0.941i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (3.48 - 10.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.96 + 6.86i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.79 - 8.61i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.35358778101275933111693259966, −9.974435289004327628716571495362, −8.677454743429633990752849069667, −8.059557222054415234064792363831, −6.67156866597210631777137271648, −6.07547850967012323673418531121, −4.84579108192757983895420252701, −4.07509165733958567074415856623, −2.69176810845821926821850226441, −0.78831146308015814067850852494, 1.90528593407225647632955287549, 3.55781973657629240887016028646, 4.48902053481448393469153216810, 4.90850255179197958962922008995, 6.58992028911727276445851018682, 7.16905929871539127409497885693, 8.733685207535888114194639941141, 9.247907002022437402341314104748, 9.917194625587194653536207652453, 11.25251187644440309864510697819

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