Properties

Label 2-532-1.1-c1-0-4
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  + 2.69·3-s + 1.42·5-s − 7-s + 4.27·9-s − 3.27·11-s + 3.42·13-s + 3.84·15-s + 5.27·17-s − 19-s − 2.69·21-s + 0.574·23-s − 2.96·25-s + 3.42·27-s − 0.122·29-s + 2.12·31-s − 8.81·33-s − 1.42·35-s − 3.96·37-s + 9.23·39-s − 4.66·41-s − 11.9·43-s + 6.08·45-s + 3.11·47-s + 49-s + 14.2·51-s − 6.08·53-s − 4.66·55-s + ⋯
L(s)  = 1  + 1.55·3-s + 0.637·5-s − 0.377·7-s + 1.42·9-s − 0.986·11-s + 0.950·13-s + 0.992·15-s + 1.27·17-s − 0.229·19-s − 0.588·21-s + 0.119·23-s − 0.593·25-s + 0.659·27-s − 0.0227·29-s + 0.381·31-s − 1.53·33-s − 0.241·35-s − 0.652·37-s + 1.47·39-s − 0.728·41-s − 1.81·43-s + 0.907·45-s + 0.454·47-s + 0.142·49-s + 1.98·51-s − 0.836·53-s − 0.628·55-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

\(L(1)\) \(\approx\) \(2.501121753\)
\(L(\frac12)\) \(\approx\) \(2.501121753\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.69T + 3T^{2} \)
5 \( 1 - 1.42T + 5T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 - 3.42T + 13T^{2} \)
17 \( 1 - 5.27T + 17T^{2} \)
23 \( 1 - 0.574T + 23T^{2} \)
29 \( 1 + 0.122T + 29T^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 + 3.96T + 37T^{2} \)
41 \( 1 + 4.66T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 - 3.11T + 47T^{2} \)
53 \( 1 + 6.08T + 53T^{2} \)
59 \( 1 + 1.72T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 5.27T + 67T^{2} \)
71 \( 1 + 6.81T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 5.23T + 83T^{2} \)
89 \( 1 + 1.45T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−10.37195765035376026449298225242, −9.949763854125872228922766102331, −8.983681525750068249055621304982, −8.255644367519764528697889914668, −7.53006651195627836331587804110, −6.30796350637534742595841829157, −5.22157330715317075106525899720, −3.69577256103269503615127567635, −2.93622564705540637267829896199, −1.74947587594220826752490724642, 1.74947587594220826752490724642, 2.93622564705540637267829896199, 3.69577256103269503615127567635, 5.22157330715317075106525899720, 6.30796350637534742595841829157, 7.53006651195627836331587804110, 8.255644367519764528697889914668, 8.983681525750068249055621304982, 9.949763854125872228922766102331, 10.37195765035376026449298225242

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