Properties

Label 2-532-1.1-c1-0-0
Degree $2$
Conductor $532$
Sign $1$
Analytic cond. $4.24804$
Root an. cond. $2.06107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.51·3-s − 2.85·5-s − 7-s + 3.34·9-s − 2.34·11-s − 0.859·13-s + 7.20·15-s + 4.34·17-s − 19-s + 2.51·21-s + 4.85·23-s + 3.17·25-s − 0.859·27-s + 9.37·29-s − 7.37·31-s + 5.89·33-s + 2.85·35-s + 2.17·37-s + 2.16·39-s + 6.69·41-s + 0.353·43-s − 9.55·45-s + 5.54·47-s + 49-s − 10.9·51-s + 9.55·53-s + 6.69·55-s + ⋯
L(s)  = 1  − 1.45·3-s − 1.27·5-s − 0.377·7-s + 1.11·9-s − 0.705·11-s − 0.238·13-s + 1.85·15-s + 1.05·17-s − 0.229·19-s + 0.549·21-s + 1.01·23-s + 0.635·25-s − 0.165·27-s + 1.74·29-s − 1.32·31-s + 1.02·33-s + 0.483·35-s + 0.357·37-s + 0.346·39-s + 1.04·41-s + 0.0539·43-s − 1.42·45-s + 0.808·47-s + 0.142·49-s − 1.53·51-s + 1.31·53-s + 0.902·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(532\)    =    \(2^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(4.24804\)
Root analytic conductor: \(2.06107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 532,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5076427729\)
\(L(\frac12)\) \(\approx\) \(0.5076427729\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
19 \( 1 + T \)
good3 \( 1 + 2.51T + 3T^{2} \)
5 \( 1 + 2.85T + 5T^{2} \)
11 \( 1 + 2.34T + 11T^{2} \)
13 \( 1 + 0.859T + 13T^{2} \)
17 \( 1 - 4.34T + 17T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 - 9.37T + 29T^{2} \)
31 \( 1 + 7.37T + 31T^{2} \)
37 \( 1 - 2.17T + 37T^{2} \)
41 \( 1 - 6.69T + 41T^{2} \)
43 \( 1 - 0.353T + 43T^{2} \)
47 \( 1 - 5.54T + 47T^{2} \)
53 \( 1 - 9.55T + 53T^{2} \)
59 \( 1 + 14.5T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + 4.34T + 67T^{2} \)
71 \( 1 - 7.89T + 71T^{2} \)
73 \( 1 - 2.23T + 73T^{2} \)
79 \( 1 - 7.36T + 79T^{2} \)
83 \( 1 + 1.83T + 83T^{2} \)
89 \( 1 + 3.31T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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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

−10.83453430069885244342201157898, −10.40833942059511545522890455696, −9.123785993389711056716324861753, −7.87876783057187559915424389150, −7.22620686671880440743459859965, −6.17080486661833610394981287929, −5.23112615924531473989453653101, −4.35626440572185263704594242378, −3.07824342109271019014623291957, −0.67232681957611485259072177045, 0.67232681957611485259072177045, 3.07824342109271019014623291957, 4.35626440572185263704594242378, 5.23112615924531473989453653101, 6.17080486661833610394981287929, 7.22620686671880440743459859965, 7.87876783057187559915424389150, 9.123785993389711056716324861753, 10.40833942059511545522890455696, 10.83453430069885244342201157898

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