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Properties

Label 2-539-7.2-c1-0-5
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  + (−0.5 + 0.866i)2-s + (1 + 1.73i)3-s + (0.500 + 0.866i)4-s + (−1 + 1.73i)5-s − 1.99·6-s − 3·8-s + (−0.499 + 0.866i)9-s + (−0.999 − 1.73i)10-s + (−0.5 − 0.866i)11-s + (−0.999 + 1.73i)12-s − 4·13-s − 3.99·15-s + (0.500 − 0.866i)16-s + (2 + 3.46i)17-s + (−0.499 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.577 + 0.999i)3-s + (0.250 + 0.433i)4-s + (−0.447 + 0.774i)5-s − 0.816·6-s − 1.06·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (−0.150 − 0.261i)11-s + (−0.288 + 0.499i)12-s − 1.10·13-s − 1.03·15-s + (0.125 − 0.216i)16-s + (0.485 + 0.840i)17-s + (−0.117 − 0.204i)18-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: $\chi_{539} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148273 - 1.16353i\)
\(L(\frac12)\) \(\approx\) \(0.148273 - 1.16353i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1 - 1.73i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (4 + 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−11.01399970901142558658085411459, −10.35417603145711302199356990572, −9.360059878873926623215291788329, −8.654573274737109421972609295479, −7.69088727673016874897390417370, −7.05166387642353150669127418995, −5.95800371243950598455606320016, −4.53986068225512843222698464083, −3.43296212852972034518347041345, −2.75367979119873177476454494544, 0.69879789330897275188748878624, 1.98172644318346283194013118055, 2.92424844745527551945513742561, 4.62513542852633258320781217837, 5.69366575655268010007718462092, 7.04171009416625584697534537626, 7.65390531723111661333422869106, 8.604315995159997365852291002654, 9.531001684614036680414375611918, 10.16352196279216803423384725093

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