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Properties

Label 2-539-77.53-c1-0-28
Degree $2$
Conductor $539$
Sign $0.798 + 0.602i$
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  + (2.31 − 0.491i)2-s + (3.27 − 1.45i)4-s + (3.03 − 2.20i)8-s + (2.93 − 0.623i)9-s + (−1.40 − 3.00i)11-s + (1.14 − 1.26i)16-s + (6.47 − 2.88i)18-s + (−4.72 − 6.25i)22-s + (4.79 + 8.29i)23-s + (0.522 + 4.97i)25-s + (−8.64 − 6.27i)29-s + (−1.74 + 3.01i)32-s + (8.71 − 6.32i)36-s + (0.142 − 1.35i)37-s − 8.74·43-s + (−8.99 − 7.79i)44-s + ⋯
L(s)  = 1  + (1.63 − 0.347i)2-s + (1.63 − 0.729i)4-s + (1.07 − 0.780i)8-s + (0.978 − 0.207i)9-s + (−0.423 − 0.905i)11-s + (0.285 − 0.316i)16-s + (1.52 − 0.679i)18-s + (−1.00 − 1.33i)22-s + (0.999 + 1.73i)23-s + (0.104 + 0.994i)25-s + (−1.60 − 1.16i)29-s + (−0.308 + 0.533i)32-s + (1.45 − 1.05i)36-s + (0.0234 − 0.223i)37-s − 1.33·43-s + (−1.35 − 1.17i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.798 + 0.602i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.798 + 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.39866 - 1.13814i\)
\(L(\frac12)\) \(\approx\) \(3.39866 - 1.13814i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.40 + 3.00i)T \)
good2 \( 1 + (-2.31 + 0.491i)T + (1.82 - 0.813i)T^{2} \)
3 \( 1 + (-2.93 + 0.623i)T^{2} \)
5 \( 1 + (-0.522 - 4.97i)T^{2} \)
13 \( 1 + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (15.5 + 6.91i)T^{2} \)
19 \( 1 + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (-4.79 - 8.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.64 + 6.27i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.24 + 30.8i)T^{2} \)
37 \( 1 + (-0.142 + 1.35i)T + (-36.1 - 7.69i)T^{2} \)
41 \( 1 + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.74T + 43T^{2} \)
47 \( 1 + (31.4 + 34.9i)T^{2} \)
53 \( 1 + (8.80 + 9.77i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-6.37 - 60.6i)T^{2} \)
67 \( 1 + (8.16 - 14.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.08 - 9.48i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (-12.3 + 2.62i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-78.4 - 57.0i)T^{2} \)
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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

−11.32472415804376071801666180232, −10.09857294229976289378755810245, −9.141156057860127748865906697923, −7.71593898277032045593480678688, −6.83925824624324887936736964555, −5.74330584756045879771058222896, −5.09015781525578167543384548314, −3.88294032878330146677028760003, −3.19783561144920957210642959694, −1.69324139725335289407341944787, 2.08304532197274915579986994761, 3.34106742018104361442946514848, 4.59336074442296142616414608002, 4.91981299861570385706600039144, 6.27979573420136782191518141805, 7.00121915392952688243484686289, 7.77780443727959812244861150496, 9.167263307830977363439417370384, 10.31581024558088232232795262007, 11.04621464870441472568369858698

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