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Properties

Label 2-539-11.4-c1-0-13
Degree $2$
Conductor $539$
Sign $0.436 + 0.899i$
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  + (−1.60 − 1.16i)2-s + (−0.861 + 2.65i)3-s + (0.598 + 1.84i)4-s + (−0.0217 + 0.0158i)5-s + (4.47 − 3.25i)6-s + (−0.0378 + 0.116i)8-s + (−3.85 − 2.80i)9-s + 0.0534·10-s + (−1.37 − 3.01i)11-s − 5.40·12-s + (−3.94 − 2.86i)13-s + (−0.0231 − 0.0713i)15-s + (3.33 − 2.42i)16-s + (1.35 − 0.986i)17-s + (2.92 + 8.99i)18-s + (−0.424 + 1.30i)19-s + ⋯
L(s)  = 1  + (−1.13 − 0.824i)2-s + (−0.497 + 1.52i)3-s + (0.299 + 0.921i)4-s + (−0.00974 + 0.00707i)5-s + (1.82 − 1.32i)6-s + (−0.0133 + 0.0411i)8-s + (−1.28 − 0.933i)9-s + 0.0168·10-s + (−0.414 − 0.909i)11-s − 1.55·12-s + (−1.09 − 0.795i)13-s + (−0.00598 − 0.0184i)15-s + (0.833 − 0.605i)16-s + (0.329 − 0.239i)17-s + (0.688 + 2.11i)18-s + (−0.0974 + 0.299i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.436 + 0.899i$
Analytic conductor: \(4.30393\)
Root analytic conductor: \(2.07459\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{539} (246, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :1/2),\ 0.436 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376786 - 0.235994i\)
\(L(\frac12)\) \(\approx\) \(0.376786 - 0.235994i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 + 3.01i)T \)
good2 \( 1 + (1.60 + 1.16i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.861 - 2.65i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (0.0217 - 0.0158i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (3.94 + 2.86i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.35 + 0.986i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.424 - 1.30i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.06T + 23T^{2} \)
29 \( 1 + (1.97 + 6.08i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.24 - 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.161 - 0.495i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.27 + 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 + (-2.76 + 8.52i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.20 - 2.32i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.00 + 9.25i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.85 + 4.97i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 + (1.65 - 1.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.23 - 9.95i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.73 + 3.44i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.10 + 1.53i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 1.21T + 89T^{2} \)
97 \( 1 + (-2.74 - 1.99i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.55916196263275357360896550332, −9.898829819317853707762492992237, −9.273921415707218388191079446361, −8.445465989814168981018568931025, −7.40446823802842217972821324869, −5.60621484515772743918911619812, −5.12405624065188786409665904605, −3.62881246004755737810723711240, −2.67148596569565533538040333206, −0.47582076771629104849641108522, 1.11272123431860948947681967848, 2.48186429369353699775833903184, 4.72450179373558934882062615667, 5.97639631148561423385465464260, 6.87557375092399060769958527377, 7.30568454167827014451711097554, 7.976672150710387090709902583508, 9.027165833786101069304349679351, 9.841637995375824796355462190481, 10.89512702264751549031748968200

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