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Properties

Label 2-539-11.3-c1-0-33
Degree $2$
Conductor $539$
Sign $-0.343 - 0.939i$
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  + (0.727 − 0.528i)2-s + (−1.00 − 3.10i)3-s + (−0.368 + 1.13i)4-s + (−1.09 − 0.794i)5-s + (−2.37 − 1.72i)6-s + (0.886 + 2.72i)8-s + (−6.19 + 4.49i)9-s − 1.21·10-s + (−1.21 − 3.08i)11-s + 3.88·12-s + (−5.18 + 3.76i)13-s + (−1.36 + 4.19i)15-s + (0.160 + 0.116i)16-s + (−0.814 − 0.591i)17-s + (−2.12 + 6.54i)18-s + (0.411 + 1.26i)19-s + ⋯
L(s)  = 1  + (0.514 − 0.373i)2-s + (−0.582 − 1.79i)3-s + (−0.184 + 0.566i)4-s + (−0.489 − 0.355i)5-s + (−0.969 − 0.704i)6-s + (0.313 + 0.965i)8-s + (−2.06 + 1.49i)9-s − 0.384·10-s + (−0.365 − 0.930i)11-s + 1.12·12-s + (−1.43 + 1.04i)13-s + (−0.352 + 1.08i)15-s + (0.0401 + 0.0291i)16-s + (−0.197 − 0.143i)17-s + (−0.501 + 1.54i)18-s + (0.0942 + 0.290i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ -0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106222 + 0.151981i\)
\(L(\frac12)\) \(\approx\) \(0.106222 + 0.151981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 + 3.08i)T \)
good2 \( 1 + (-0.727 + 0.528i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (1.00 + 3.10i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.09 + 0.794i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (5.18 - 3.76i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.814 + 0.591i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.411 - 1.26i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.61T + 23T^{2} \)
29 \( 1 + (-0.441 + 1.35i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.74 - 1.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.32 - 4.07i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.09 + 6.43i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.03T + 43T^{2} \)
47 \( 1 + (1.60 + 4.92i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.10 - 0.800i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.94 - 9.07i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.62 + 4.08i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 + (-3.21 - 2.33i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.19 + 9.82i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-10.9 + 7.95i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.32 + 2.41i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.94T + 89T^{2} \)
97 \( 1 + (-1.39 + 1.01i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.71134347545868712044033344396, −8.970191421609844296261226270771, −8.151004523589156184444109202661, −7.48644713526831314302999661689, −6.68446007592729620613276428437, −5.48083005567486188986880546342, −4.59022795693414553057666891421, −3.00748771557660752912532099117, −1.93918219055204052088709348460, −0.092690856474828052927021239547, 3.03256175514371735951968119954, 4.16679955525461756239599414898, 4.98784981060724815905375629505, 5.40275158937461315981115047666, 6.65492616844407091607932655842, 7.72573156834689004520186496052, 9.272214000108996533599716338962, 9.788259049735384090900747117051, 10.50014031280641113675374986455, 11.09612902869024799417100171039

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