Properties

Label 2-532-1.1-c1-0-3
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  − 1.17·3-s + 3.43·5-s − 7-s − 1.61·9-s + 2.61·11-s + 5.43·13-s − 4.04·15-s − 0.611·17-s − 19-s + 1.17·21-s − 1.43·23-s + 6.79·25-s + 5.43·27-s + 1.74·29-s + 0.255·31-s − 3.07·33-s − 3.43·35-s + 5.79·37-s − 6.40·39-s + 8.96·41-s + 7.58·43-s − 5.53·45-s − 10.6·47-s + 49-s + 0.720·51-s + 5.53·53-s + 8.96·55-s + ⋯
L(s)  = 1  − 0.680·3-s + 1.53·5-s − 0.377·7-s − 0.537·9-s + 0.787·11-s + 1.50·13-s − 1.04·15-s − 0.148·17-s − 0.229·19-s + 0.257·21-s − 0.298·23-s + 1.35·25-s + 1.04·27-s + 0.323·29-s + 0.0458·31-s − 0.535·33-s − 0.580·35-s + 0.951·37-s − 1.02·39-s + 1.40·41-s + 1.15·43-s − 0.825·45-s − 1.55·47-s + 0.142·49-s + 0.100·51-s + 0.760·53-s + 1.20·55-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

\(L(1)\) \(\approx\) \(1.493271048\)
\(L(\frac12)\) \(\approx\) \(1.493271048\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.17T + 3T^{2} \)
5 \( 1 - 3.43T + 5T^{2} \)
11 \( 1 - 2.61T + 11T^{2} \)
13 \( 1 - 5.43T + 13T^{2} \)
17 \( 1 + 0.611T + 17T^{2} \)
23 \( 1 + 1.43T + 23T^{2} \)
29 \( 1 - 1.74T + 29T^{2} \)
31 \( 1 - 0.255T + 31T^{2} \)
37 \( 1 - 5.79T + 37T^{2} \)
41 \( 1 - 8.96T + 41T^{2} \)
43 \( 1 - 7.58T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 5.53T + 53T^{2} \)
59 \( 1 - 4.30T + 59T^{2} \)
61 \( 1 + 1.27T + 61T^{2} \)
67 \( 1 - 0.611T + 67T^{2} \)
71 \( 1 + 1.07T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 7.88T + 97T^{2} \)
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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

−10.88572931534637153216550906771, −9.971613477149009093267314267493, −9.173684892032731244241915386119, −8.422758858790219811209059391082, −6.78165107694721299597808291444, −6.02292060836695184786515810754, −5.70251231015803247666193158640, −4.22143293068117386592167793388, −2.74825576966717284429858589035, −1.27823511515292654918125295076, 1.27823511515292654918125295076, 2.74825576966717284429858589035, 4.22143293068117386592167793388, 5.70251231015803247666193158640, 6.02292060836695184786515810754, 6.78165107694721299597808291444, 8.422758858790219811209059391082, 9.173684892032731244241915386119, 9.971613477149009093267314267493, 10.88572931534637153216550906771

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