Properties

Label 2-532-1.1-c1-0-5
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  + 1.79·3-s + 3·5-s + 7-s + 0.208·9-s + 0.791·11-s − 13-s + 5.37·15-s − 0.791·17-s + 19-s + 1.79·21-s − 4.58·23-s + 4·25-s − 5.00·27-s − 0.791·29-s − 6.37·31-s + 1.41·33-s + 3·35-s + 5·37-s − 1.79·39-s + 0.791·41-s + 2·43-s + 0.626·45-s + 1.41·47-s + 49-s − 1.41·51-s + 5.37·53-s + 2.37·55-s + ⋯
L(s)  = 1  + 1.03·3-s + 1.34·5-s + 0.377·7-s + 0.0695·9-s + 0.238·11-s − 0.277·13-s + 1.38·15-s − 0.191·17-s + 0.229·19-s + 0.390·21-s − 0.955·23-s + 0.800·25-s − 0.962·27-s − 0.146·29-s − 1.14·31-s + 0.246·33-s + 0.507·35-s + 0.821·37-s − 0.286·39-s + 0.123·41-s + 0.304·43-s + 0.0933·45-s + 0.206·47-s + 0.142·49-s − 0.198·51-s + 0.738·53-s + 0.320·55-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

\(L(1)\) \(\approx\) \(2.418599729\)
\(L(\frac12)\) \(\approx\) \(2.418599729\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.79T + 3T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
11 \( 1 - 0.791T + 11T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + 0.791T + 17T^{2} \)
23 \( 1 + 4.58T + 23T^{2} \)
29 \( 1 + 0.791T + 29T^{2} \)
31 \( 1 + 6.37T + 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 0.791T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 - 1.41T + 47T^{2} \)
53 \( 1 - 5.37T + 53T^{2} \)
59 \( 1 + 6.16T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 4.37T + 67T^{2} \)
71 \( 1 - 6.16T + 71T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 0.626T + 83T^{2} \)
89 \( 1 + 1.58T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.65956759752428658695631982286, −9.671369471696962735996023165166, −9.209891857393727686192046214456, −8.291756036876013472680338059872, −7.38138449244145568345351900091, −6.15569878512594788467150208518, −5.34489458742315857556074066981, −3.96682945915399810954508145916, −2.64136758039257950310911705616, −1.79012099402188512581923391590, 1.79012099402188512581923391590, 2.64136758039257950310911705616, 3.96682945915399810954508145916, 5.34489458742315857556074066981, 6.15569878512594788467150208518, 7.38138449244145568345351900091, 8.291756036876013472680338059872, 9.209891857393727686192046214456, 9.671369471696962735996023165166, 10.65956759752428658695631982286

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