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Properties

Label 2-539-7.4-c1-0-27
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 + (1.61 − 2.80i)3-s + (−1.5 + 2.59i)4-s + (−1 − 1.73i)5-s + 7.23·6-s − 2.23·8-s + (−3.73 − 6.47i)9-s + (2.23 − 3.87i)10-s + (0.5 − 0.866i)11-s + (4.85 + 8.40i)12-s + 1.23·13-s − 6.47·15-s + (0.499 + 0.866i)16-s + (0.618 − 1.07i)17-s + (8.35 − 14.4i)18-s + (−1.23 − 2.14i)19-s + ⋯
L(s)  = 1  + (0.790 + 1.36i)2-s + (0.934 − 1.61i)3-s + (−0.750 + 1.29i)4-s + (−0.447 − 0.774i)5-s + 2.95·6-s − 0.790·8-s + (−1.24 − 2.15i)9-s + (0.707 − 1.22i)10-s + (0.150 − 0.261i)11-s + (1.40 + 2.42i)12-s + 0.342·13-s − 1.67·15-s + (0.124 + 0.216i)16-s + (0.149 − 0.259i)17-s + (1.96 − 3.41i)18-s + (−0.283 − 0.491i)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: $\chi_{539} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60857 - 0.332419i\)
\(L(\frac12)\) \(\approx\) \(2.60857 - 0.332419i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 2.80i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.23T + 13T^{2} \)
17 \( 1 + (-0.618 + 1.07i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.23 + 2.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.23 - 5.60i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.472T + 29T^{2} \)
31 \( 1 + (3.61 - 6.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.236 + 0.408i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.76T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (-3.61 - 6.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.23 - 7.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.61 + 2.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.38 + 2.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.76 - 4.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.52T + 71T^{2} \)
73 \( 1 + (2.61 - 4.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.47 + 7.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−11.14887764763805061681379492242, −9.091893307529785646207501466920, −8.683801603802681574494493754488, −7.71036786314683447360842044892, −7.30483261970378141121782912935, −6.37211565424609510374812786034, −5.50182381850062081084515544476, −4.20184981005406760522424458129, −3.02569756970455851140797070376, −1.24448746296506385756608264342, 2.28753809640506010224077405796, 3.16219838229870067734012668244, 3.93020745287735543370333328026, 4.52921055138864951451262564194, 5.71281609949171338462861166550, 7.45280500634263382231823822859, 8.586644682909052661819096945608, 9.489449279402143880436402760351, 10.24820510057332239440154432933, 10.88613884717147105595813944644

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