Properties

Label 2-539-77.9-c1-0-24
Degree $2$
Conductor $539$
Sign $0.911 - 0.411i$
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  + (0.448 + 0.497i)2-s + (2.86 + 1.27i)3-s + (0.162 − 1.54i)4-s + (−2.09 − 0.445i)5-s + (0.649 + 1.99i)6-s + (1.92 − 1.39i)8-s + (4.58 + 5.09i)9-s + (−0.717 − 1.24i)10-s + (3.25 − 0.628i)11-s + (2.43 − 4.21i)12-s + (0.781 − 2.40i)13-s + (−5.44 − 3.95i)15-s + (−1.47 − 0.313i)16-s + (−1.19 + 1.33i)17-s + (−0.480 + 4.56i)18-s + (0.703 + 6.69i)19-s + ⋯
L(s)  = 1  + (0.316 + 0.351i)2-s + (1.65 + 0.737i)3-s + (0.0810 − 0.771i)4-s + (−0.937 − 0.199i)5-s + (0.265 + 0.816i)6-s + (0.680 − 0.494i)8-s + (1.52 + 1.69i)9-s + (−0.227 − 0.393i)10-s + (0.981 − 0.189i)11-s + (0.703 − 1.21i)12-s + (0.216 − 0.666i)13-s + (−1.40 − 1.02i)15-s + (−0.369 − 0.0784i)16-s + (−0.290 + 0.322i)17-s + (−0.113 + 1.07i)18-s + (0.161 + 1.53i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64202 + 0.568199i\)
\(L(\frac12)\) \(\approx\) \(2.64202 + 0.568199i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.25 + 0.628i)T \)
good2 \( 1 + (-0.448 - 0.497i)T + (-0.209 + 1.98i)T^{2} \)
3 \( 1 + (-2.86 - 1.27i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (2.09 + 0.445i)T + (4.56 + 2.03i)T^{2} \)
13 \( 1 + (-0.781 + 2.40i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.19 - 1.33i)T + (-1.77 - 16.9i)T^{2} \)
19 \( 1 + (-0.703 - 6.69i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-1.58 + 2.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.747 - 0.543i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.93 - 0.624i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (1.37 - 0.613i)T + (24.7 - 27.4i)T^{2} \)
41 \( 1 + (4.49 - 3.26i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.42T + 43T^{2} \)
47 \( 1 + (0.459 + 4.37i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (0.652 - 0.138i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (-0.0385 + 0.366i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (4.90 + 1.04i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-0.451 - 0.781i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.59 + 14.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.840 - 7.99i)T + (-71.4 - 15.1i)T^{2} \)
79 \( 1 + (-2.71 - 3.01i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (1.25 + 3.85i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-4.15 + 7.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.63 + 8.09i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.52534472109192668306550810294, −9.981580824510661995460668722399, −8.980548117533406979811859953409, −8.287382800461282888389867000585, −7.52424352918975392425892788103, −6.34282289138252700376448226467, −4.97448710640903154733554568067, −4.01484316121663469413412539722, −3.39290268614565612271900514701, −1.68228235154610373598169409904, 1.77421774176279048575968266525, 2.95405215586104184411138098233, 3.68539111345569506459767777503, 4.49238927503320996739354864967, 6.90759232518335328377803406914, 7.14119692035164259110912583772, 8.107642597209567037092796017397, 8.849103272859573908480345573913, 9.456590743488329304878400841842, 11.16585331261293993662946033295

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