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Properties

Label 2-539-11.5-c1-0-19
Degree $2$
Conductor $539$
Sign $0.603 - 0.797i$
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  + (0.303 + 0.932i)2-s + (1.59 + 1.16i)3-s + (0.839 − 0.610i)4-s + (0.516 − 1.58i)5-s + (−0.598 + 1.84i)6-s + (2.41 + 1.75i)8-s + (0.279 + 0.859i)9-s + 1.63·10-s + (−0.943 + 3.17i)11-s + 2.05·12-s + (−0.657 − 2.02i)13-s + (2.67 − 1.94i)15-s + (−0.261 + 0.805i)16-s + (−1.22 + 3.76i)17-s + (−0.716 + 0.520i)18-s + (−3.19 − 2.31i)19-s + ⋯
L(s)  = 1  + (0.214 + 0.659i)2-s + (0.922 + 0.670i)3-s + (0.419 − 0.305i)4-s + (0.231 − 0.711i)5-s + (−0.244 + 0.752i)6-s + (0.852 + 0.619i)8-s + (0.0930 + 0.286i)9-s + 0.518·10-s + (−0.284 + 0.958i)11-s + 0.591·12-s + (−0.182 − 0.561i)13-s + (0.689 − 0.501i)15-s + (−0.0654 + 0.201i)16-s + (−0.296 + 0.913i)17-s + (−0.168 + 0.122i)18-s + (−0.731 − 0.531i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.603 - 0.797i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.603 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.28033 + 1.13331i\)
\(L(\frac12)\) \(\approx\) \(2.28033 + 1.13331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.943 - 3.17i)T \)
good2 \( 1 + (-0.303 - 0.932i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.59 - 1.16i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.516 + 1.58i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.657 + 2.02i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.22 - 3.76i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.19 + 2.31i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.10T + 23T^{2} \)
29 \( 1 + (0.200 - 0.145i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.186 - 0.575i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.84 - 1.33i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.34 + 3.15i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (10.5 + 7.65i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.208 + 0.642i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.11 - 0.811i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.46 - 10.6i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + (0.563 - 1.73i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.10 + 2.25i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.77 + 14.7i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.45 + 7.55i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.17T + 89T^{2} \)
97 \( 1 + (-3.71 - 11.4i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.59306784160592335096258830116, −10.08808908715168543880305127420, −8.963126201614488160629755008212, −8.440217334053147795265658622781, −7.32315465848380084449205807921, −6.44292515735818950430404366681, −5.14141574103532261127916708016, −4.60430616004814871840449536970, −3.11914735292749111342633321473, −1.80154026728981185871721210100, 1.68555052185815589054470191833, 2.80422544561592277852632917680, 3.25153863459221074010074015958, 4.82352989558077309592007717731, 6.49520640629838663911317886868, 7.05243141359763152210151901875, 8.010045680380978492626988569256, 8.804013706741760088541597609024, 9.952177284381143312803300890782, 10.98928504336931793415645936753

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