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Properties

Label 2-539-11.5-c1-0-25
Degree $2$
Conductor $539$
Sign $-0.158 + 0.987i$
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  + (−0.240 − 0.739i)2-s + (0.5 + 0.363i)3-s + (1.12 − 0.820i)4-s + (0.0687 − 0.211i)5-s + (0.148 − 0.456i)6-s + (−2.13 − 1.55i)8-s + (−0.809 − 2.48i)9-s − 0.173·10-s + (0.660 + 3.25i)11-s + 0.862·12-s + (−2.01 − 6.20i)13-s + (0.111 − 0.0808i)15-s + (0.228 − 0.702i)16-s + (1.33 − 4.12i)17-s + (−1.64 + 1.19i)18-s + (2.35 + 1.71i)19-s + ⋯
L(s)  = 1  + (−0.169 − 0.522i)2-s + (0.288 + 0.209i)3-s + (0.564 − 0.410i)4-s + (0.0307 − 0.0946i)5-s + (0.0606 − 0.186i)6-s + (−0.755 − 0.548i)8-s + (−0.269 − 0.829i)9-s − 0.0547·10-s + (0.199 + 0.979i)11-s + 0.248·12-s + (−0.559 − 1.72i)13-s + (0.0287 − 0.0208i)15-s + (0.0570 − 0.175i)16-s + (0.324 − 0.999i)17-s + (−0.388 + 0.282i)18-s + (0.541 + 0.393i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $-0.158 + 0.987i$
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: \((2,\ 539,\ (\ :1/2),\ -0.158 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00212 - 1.17612i\)
\(L(\frac12)\) \(\approx\) \(1.00212 - 1.17612i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.660 - 3.25i)T \)
good2 \( 1 + (0.240 + 0.739i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.5 - 0.363i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.0687 + 0.211i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (2.01 + 6.20i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.33 + 4.12i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.35 - 1.71i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.89T + 23T^{2} \)
29 \( 1 + (-3.05 + 2.21i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.12 - 6.55i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.57 + 3.32i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.08 - 0.786i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + (-4.89 - 3.55i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.530 + 1.63i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.71 - 5.60i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.97 + 9.15i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 1.27T + 67T^{2} \)
71 \( 1 + (2.87 - 8.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.52 + 3.28i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.39 + 4.30i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.48 - 10.7i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 7.92T + 89T^{2} \)
97 \( 1 + (-2.79 - 8.61i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.32764811515170939463525447856, −9.880546245419073228159986702073, −9.145884386990171650889872019855, −7.88484555399070437620720782413, −6.99955569374792513231999125438, −5.93031875856113801188953798168, −4.94735556317273193471270184599, −3.38774044461169239021210489690, −2.59913951390149605061299916573, −0.932541764103380128210815072107, 1.98914702687462204010289449255, 3.05458952335679835689694500721, 4.43295677907406097165640776257, 5.83650306146955785627379713034, 6.60457587638374594994469068258, 7.50790615179725377208061486007, 8.324751086737682012588125147821, 8.944437011547494841884538076918, 10.17910548486575457672268515060, 11.26137728573074164052974321486

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