Properties

Label 2-539-77.25-c1-0-5
Degree $2$
Conductor $539$
Sign $0.849 - 0.528i$
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  + (−1.33 − 0.596i)2-s + (−1.58 − 0.336i)3-s + (0.101 + 0.112i)4-s + (0.0487 − 0.464i)5-s + (1.91 + 1.39i)6-s + (0.837 + 2.57i)8-s + (−0.348 − 0.155i)9-s + (−0.342 + 0.592i)10-s + (−1.01 − 3.15i)11-s + (−0.122 − 0.212i)12-s + (−1.28 + 0.930i)13-s + (−0.233 + 0.718i)15-s + (0.447 − 4.25i)16-s + (−4.77 + 2.12i)17-s + (0.374 + 0.416i)18-s + (−2.82 + 3.13i)19-s + ⋯
L(s)  = 1  + (−0.947 − 0.421i)2-s + (−0.913 − 0.194i)3-s + (0.0506 + 0.0562i)4-s + (0.0218 − 0.207i)5-s + (0.783 + 0.569i)6-s + (0.296 + 0.911i)8-s + (−0.116 − 0.0517i)9-s + (−0.108 + 0.187i)10-s + (−0.305 − 0.952i)11-s + (−0.0353 − 0.0612i)12-s + (−0.355 + 0.257i)13-s + (−0.0602 + 0.185i)15-s + (0.111 − 1.06i)16-s + (−1.15 + 0.515i)17-s + (0.0883 + 0.0981i)18-s + (−0.648 + 0.719i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.849 - 0.528i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (410, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.849 - 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.297658 + 0.0850333i\)
\(L(\frac12)\) \(\approx\) \(0.297658 + 0.0850333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 + 3.15i)T \)
good2 \( 1 + (1.33 + 0.596i)T + (1.33 + 1.48i)T^{2} \)
3 \( 1 + (1.58 + 0.336i)T + (2.74 + 1.22i)T^{2} \)
5 \( 1 + (-0.0487 + 0.464i)T + (-4.89 - 1.03i)T^{2} \)
13 \( 1 + (1.28 - 0.930i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.77 - 2.12i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (2.82 - 3.13i)T + (-1.98 - 18.8i)T^{2} \)
23 \( 1 + (-0.902 - 1.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.840 + 2.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.135 - 1.28i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (-1.89 + 0.403i)T + (33.8 - 15.0i)T^{2} \)
41 \( 1 + (-0.321 - 0.990i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-4.27 + 4.75i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (-1.38 - 13.1i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (-5.75 - 6.39i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (1.59 - 15.1i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (-2.33 + 4.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.88 + 5.72i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-8.91 - 9.89i)T + (-7.63 + 72.6i)T^{2} \)
79 \( 1 + (-3.27 - 1.45i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (13.9 + 10.1i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.45 + 7.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.18 - 1.58i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.82776032406270726999934866990, −10.26783162451486520092011859394, −8.936451968558033137002235668032, −8.660388836089690205158268687543, −7.43425628878038947987603203311, −6.19562949019071999958562948239, −5.50689243256249660364464948533, −4.35618635584215772914311259386, −2.55961237812147282896374830178, −1.01368648177773341261312339192, 0.34845751108907971996080621836, 2.53003733348728895260325066395, 4.35412905565509029488877408311, 5.11298079328257739223435695016, 6.53771142926478992258016631042, 7.02480507147586081712366316173, 8.122931351997597941546729009967, 8.990438424402794258927442306319, 9.822549942798765879764340064396, 10.69803020838931364832851478184

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