Properties

Label 2-539-49.44-c1-0-8
Degree $2$
Conductor $539$
Sign $-0.892 - 0.451i$
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  + (−2.45 + 0.370i)2-s + (2.28 + 1.55i)3-s + (3.98 − 1.22i)4-s + (−0.252 + 3.36i)5-s + (−6.17 − 2.97i)6-s + (−0.782 − 2.52i)7-s + (−4.85 + 2.33i)8-s + (1.68 + 4.29i)9-s + (−0.626 − 8.36i)10-s + (0.365 − 0.930i)11-s + (10.9 + 3.39i)12-s + (0.0347 + 0.0436i)13-s + (2.85 + 5.91i)14-s + (−5.80 + 7.28i)15-s + (4.17 − 2.84i)16-s + (−5.17 + 4.80i)17-s + ⋯
L(s)  = 1  + (−1.73 + 0.261i)2-s + (1.31 + 0.897i)3-s + (1.99 − 0.614i)4-s + (−0.112 + 1.50i)5-s + (−2.52 − 1.21i)6-s + (−0.295 − 0.955i)7-s + (−1.71 + 0.827i)8-s + (0.562 + 1.43i)9-s + (−0.198 − 2.64i)10-s + (0.110 − 0.280i)11-s + (3.17 + 0.979i)12-s + (0.00964 + 0.0120i)13-s + (0.763 + 1.58i)14-s + (−1.49 + 1.88i)15-s + (1.04 − 0.711i)16-s + (−1.25 + 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.892 - 0.451i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185167 + 0.776422i\)
\(L(\frac12)\) \(\approx\) \(0.185167 + 0.776422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.782 + 2.52i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
good2 \( 1 + (2.45 - 0.370i)T + (1.91 - 0.589i)T^{2} \)
3 \( 1 + (-2.28 - 1.55i)T + (1.09 + 2.79i)T^{2} \)
5 \( 1 + (0.252 - 3.36i)T + (-4.94 - 0.745i)T^{2} \)
13 \( 1 + (-0.0347 - 0.0436i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (5.17 - 4.80i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (2.37 - 4.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.841 - 0.780i)T + (1.71 + 22.9i)T^{2} \)
29 \( 1 + (-1.86 - 8.17i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (0.622 + 1.07i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.96 - 1.53i)T + (30.5 + 20.8i)T^{2} \)
41 \( 1 + (-5.41 + 2.60i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.83 + 0.885i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (10.4 - 1.58i)T + (44.9 - 13.8i)T^{2} \)
53 \( 1 + (-5.56 + 1.71i)T + (43.7 - 29.8i)T^{2} \)
59 \( 1 + (0.875 + 11.6i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-10.0 - 3.10i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (3.98 + 6.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.254 - 1.11i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-9.27 - 1.39i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (6.01 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.15 - 2.69i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-4.49 - 11.4i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 - 8.05T + 97T^{2} \)
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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

−10.76149789870550650263856644494, −10.14242364482532236984932879067, −9.511081620983967073999925943117, −8.510326474965118063947120036877, −7.970987719589505133304213117764, −6.97185355452489387846976340469, −6.38822692376167773015245453089, −4.05443898710484606489126462826, −3.19238195749012116321609712609, −2.01624119713391742345818985786, 0.66812732807488373222293415370, 2.04972386258394556018317314967, 2.69498894691443829510611830037, 4.63974791851736157175957743442, 6.39306613341708897507674015786, 7.34357583478842778620609876043, 8.186174463040955091784194502152, 8.862880914532562939514975728592, 9.080296002757386292104290057681, 9.826844429480984417474568722037

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