Properties

Label 2-539-7.2-c1-0-32
Degree $2$
Conductor $539$
Sign $-0.386 - 0.922i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.386 - 0.922i$
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: \((2,\ 539,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855511 + 1.28613i\)
\(L(\frac12)\) \(\approx\) \(0.855511 + 1.28613i\)
\(L(\frac{3}{2})\) 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

−10.75889983206408956456045832462, −9.618992190932875602044419530539, −8.462754275611316461754698456907, −7.25713622877921567440570502613, −6.38424337795378804715437528780, −5.19602647207325461040749233551, −4.77388080105133914209378598766, −2.83639151232509788616126098285, −1.78356681988116215703972806111, −0.822784201335728837393333274764, 3.25711358255271692137049399901, 4.18564766118855557167742205701, 5.09241956890465159180522618181, 5.86390609135838535216053441414, 6.41713484209471017391483522649, 7.49870047209447884558607894179, 8.814686818551771929748062150013, 9.781402186885600619208762827177, 10.48314928600844495990946162747, 11.29743617563840446506212792957

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