Properties

Label 2-538-1.1-c1-0-18
Degree $2$
Conductor $538$
Sign $-1$
Analytic cond. $4.29595$
Root an. cond. $2.07266$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 0.980·3-s + 4-s − 0.685·5-s − 0.980·6-s − 1.46·7-s − 8-s − 2.03·9-s + 0.685·10-s + 0.0958·11-s + 0.980·12-s − 5.45·13-s + 1.46·14-s − 0.672·15-s + 16-s − 6.05·17-s + 2.03·18-s + 5.10·19-s − 0.685·20-s − 1.43·21-s − 0.0958·22-s − 4.13·23-s − 0.980·24-s − 4.52·25-s + 5.45·26-s − 4.94·27-s − 1.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.566·3-s + 0.5·4-s − 0.306·5-s − 0.400·6-s − 0.552·7-s − 0.353·8-s − 0.679·9-s + 0.216·10-s + 0.0289·11-s + 0.283·12-s − 1.51·13-s + 0.390·14-s − 0.173·15-s + 0.250·16-s − 1.46·17-s + 0.480·18-s + 1.17·19-s − 0.153·20-s − 0.312·21-s − 0.0204·22-s − 0.862·23-s − 0.200·24-s − 0.905·25-s + 1.06·26-s − 0.951·27-s − 0.276·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(538\)    =    \(2 \cdot 269\)
Sign: $-1$
Analytic conductor: \(4.29595\)
Root analytic conductor: \(2.07266\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 538,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
269 \( 1 + T \)
good3 \( 1 - 0.980T + 3T^{2} \)
5 \( 1 + 0.685T + 5T^{2} \)
7 \( 1 + 1.46T + 7T^{2} \)
11 \( 1 - 0.0958T + 11T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
17 \( 1 + 6.05T + 17T^{2} \)
19 \( 1 - 5.10T + 19T^{2} \)
23 \( 1 + 4.13T + 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 - 1.86T + 31T^{2} \)
37 \( 1 - 9.97T + 37T^{2} \)
41 \( 1 + 5.17T + 41T^{2} \)
43 \( 1 + 2.42T + 43T^{2} \)
47 \( 1 + 3.13T + 47T^{2} \)
53 \( 1 + 6.72T + 53T^{2} \)
59 \( 1 - 5.30T + 59T^{2} \)
61 \( 1 - 7.51T + 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 6.38T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 - 2.28T + 97T^{2} \)
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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

−9.952704404298776188549141123199, −9.625057528633788924438493198650, −8.582060928394737458020005394005, −7.85716393945938051458077367939, −6.98385039008245990505615479113, −5.95472762095435739740519685514, −4.58852126074377547156857133849, −3.16758831231193328187824031299, −2.25944495664960247894814222884, 0, 2.25944495664960247894814222884, 3.16758831231193328187824031299, 4.58852126074377547156857133849, 5.95472762095435739740519685514, 6.98385039008245990505615479113, 7.85716393945938051458077367939, 8.582060928394737458020005394005, 9.625057528633788924438493198650, 9.952704404298776188549141123199

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