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Properties

Label 2-539-7.2-c1-0-11
Degree $2$
Conductor $539$
Sign $0.968 - 0.250i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.11 + 1.93i)2-s + (−0.618 − 1.07i)3-s + (−1.5 − 2.59i)4-s + (−1 + 1.73i)5-s + 2.76·6-s + 2.23·8-s + (0.736 − 1.27i)9-s + (−2.23 − 3.87i)10-s + (0.5 + 0.866i)11-s + (−1.85 + 3.21i)12-s − 3.23·13-s + 2.47·15-s + (0.499 − 0.866i)16-s + (−1.61 − 2.80i)17-s + (1.64 + 2.85i)18-s + (3.23 − 5.60i)19-s + ⋯
L(s)  = 1  + (−0.790 + 1.36i)2-s + (−0.356 − 0.618i)3-s + (−0.750 − 1.29i)4-s + (−0.447 + 0.774i)5-s + 1.12·6-s + 0.790·8-s + (0.245 − 0.424i)9-s + (−0.707 − 1.22i)10-s + (0.150 + 0.261i)11-s + (−0.535 + 0.927i)12-s − 0.897·13-s + 0.638·15-s + (0.124 − 0.216i)16-s + (−0.392 − 0.679i)17-s + (0.387 + 0.671i)18-s + (0.742 − 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606149 + 0.0772438i\)
\(L(\frac12)\) \(\approx\) \(0.606149 + 0.0772438i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (1.11 - 1.93i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.618 + 1.07i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
17 \( 1 + (1.61 + 2.80i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.23 + 5.60i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.23 - 2.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 + (1.38 + 2.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.23 + 7.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-1.38 + 2.39i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.236 - 0.408i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.618 + 1.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.61 - 6.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.23 + 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + (0.381 + 0.661i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.47 + 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−10.75652178398619552433269526293, −9.515725425729820220733295593867, −9.136637479540768602112096570321, −7.55704003438682787183079477920, −7.39968681691492480206979434580, −6.65602461053806146121869447091, −5.76765834664645120392100170624, −4.52816663494985412323786277799, −2.78795135865811864523611655648, −0.58818803448977961567802267588, 1.14239053234515431091069104261, 2.61154015120235771882239822792, 4.00674898605575350253888151167, 4.69977730771955953903722951491, 6.00933081208038237373277553744, 7.71135507238770598415715510866, 8.389964260597151027988642685790, 9.316356502336761435909799041305, 10.10462944839272191504593724072, 10.61774787471990852516563964704

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