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Properties

Label 2-539-11.10-c0-0-1
Degree $2$
Conductor $539$
Sign $1$
Analytic cond. $0.268996$
Root an. cond. $0.518648$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 9-s + 11-s + 16-s − 2·23-s − 25-s − 36-s + 2·37-s + 44-s − 2·53-s + 64-s − 2·67-s − 2·71-s + 81-s − 2·92-s − 99-s − 100-s + 2·113-s + ⋯
L(s)  = 1  + 4-s − 9-s + 11-s + 16-s − 2·23-s − 25-s − 36-s + 2·37-s + 44-s − 2·53-s + 64-s − 2·67-s − 2·71-s + 81-s − 2·92-s − 99-s − 100-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(539\)    =    \(7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.268996\)
Root analytic conductor: \(0.518648\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{539} (197, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 539,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.056825846\)
\(L(\frac12)\) \(\approx\) \(1.056825846\)
\(L(1)\) 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 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( ( 1 - T )^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 + T )^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 + T )^{2} \)
71 \( ( 1 + T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−11.28739282962848698505120632953, −10.19909880611827908844286208940, −9.351432699561081918091144700232, −8.215578794023227119547930381184, −7.51484051789775687344911514592, −6.17907430091643420023445244776, −5.95434117066505121574594539243, −4.25036824768570974284525817518, −3.07132270284957325551048573694, −1.85005421960275431682671703240, 1.85005421960275431682671703240, 3.07132270284957325551048573694, 4.25036824768570974284525817518, 5.95434117066505121574594539243, 6.17907430091643420023445244776, 7.51484051789775687344911514592, 8.215578794023227119547930381184, 9.351432699561081918091144700232, 10.19909880611827908844286208940, 11.28739282962848698505120632953

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