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Properties

Label 2-539-11.4-c1-0-9
Degree $2$
Conductor $539$
Sign $0.963 + 0.268i$
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  + (−2.28 − 1.66i)2-s + (1.84 + 5.68i)4-s + (3.47 − 10.7i)8-s + (2.42 + 1.76i)9-s + (−3.17 − 0.964i)11-s + (−16.0 + 11.6i)16-s + (−2.61 − 8.06i)18-s + (5.65 + 7.47i)22-s + 7.50·23-s + (−1.54 + 4.75i)25-s + (1.42 + 4.37i)29-s + 33.5·32-s + (−5.54 + 17.0i)36-s + (2.53 + 7.80i)37-s + 6.59·43-s + (−0.376 − 19.8i)44-s + ⋯
L(s)  = 1  + (−1.61 − 1.17i)2-s + (0.924 + 2.84i)4-s + (1.22 − 3.78i)8-s + (0.809 + 0.587i)9-s + (−0.956 − 0.290i)11-s + (−4.00 + 2.91i)16-s + (−0.617 − 1.89i)18-s + (1.20 + 1.59i)22-s + 1.56·23-s + (−0.309 + 0.951i)25-s + (0.264 + 0.812i)29-s + 5.92·32-s + (−0.924 + 2.84i)36-s + (0.417 + 1.28i)37-s + 1.00·43-s + (−0.0567 − 2.99i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.268i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $0.963 + 0.268i$
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: \((2,\ 539,\ (\ :1/2),\ 0.963 + 0.268i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625561 - 0.0854196i\)
\(L(\frac12)\) \(\approx\) \(0.625561 - 0.0854196i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.17 + 0.964i)T \)
good2 \( 1 + (2.28 + 1.66i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.50T + 23T^{2} \)
29 \( 1 + (-1.42 - 4.37i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.53 - 7.80i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.59T + 43T^{2} \)
47 \( 1 + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.51 + 1.09i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 + (-7.95 + 5.78i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-12.7 - 9.28i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (29.9 + 92.2i)T^{2} \)
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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

−10.82223499700806973641064224008, −9.937163428445196729830757541005, −9.220215089778119085027637256610, −8.275774617031878508143489373181, −7.57331644674307782710147816441, −6.80498524154789955830612719136, −4.87391546990344693482586252953, −3.47812009637979978507729471080, −2.46551691318605730823213196142, −1.17428691856573189125309987034, 0.74918193506168451923392434774, 2.31141733878217379427440428490, 4.63466935211992110298635119771, 5.69273467201521521875372073599, 6.64027792661203150115516807053, 7.38849553224832706412583759622, 8.091591534567407804084598270531, 9.093809578506075818043263567892, 9.728194154620770224268521820741, 10.48910520443272203807034020100

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