Properties

Label 2-3549-1.1-c1-0-42
Degree $2$
Conductor $3549$
Sign $1$
Analytic cond. $28.3389$
Root an. cond. $5.32343$
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  − 0.775·2-s + 3-s − 1.39·4-s − 3.03·5-s − 0.775·6-s + 7-s + 2.63·8-s + 9-s + 2.35·10-s + 6.14·11-s − 1.39·12-s − 0.775·14-s − 3.03·15-s + 0.750·16-s + 6.44·17-s − 0.775·18-s + 4.32·19-s + 4.24·20-s + 21-s − 4.76·22-s − 8.92·23-s + 2.63·24-s + 4.20·25-s + 27-s − 1.39·28-s − 2.44·29-s + 2.35·30-s + ⋯
L(s)  = 1  − 0.548·2-s + 0.577·3-s − 0.699·4-s − 1.35·5-s − 0.316·6-s + 0.377·7-s + 0.932·8-s + 0.333·9-s + 0.744·10-s + 1.85·11-s − 0.403·12-s − 0.207·14-s − 0.783·15-s + 0.187·16-s + 1.56·17-s − 0.182·18-s + 0.991·19-s + 0.948·20-s + 0.218·21-s − 1.01·22-s − 1.86·23-s + 0.538·24-s + 0.841·25-s + 0.192·27-s − 0.264·28-s − 0.454·29-s + 0.429·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(28.3389\)
Root analytic conductor: \(5.32343\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.352263674\)
\(L(\frac12)\) \(\approx\) \(1.352263674\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + 0.775T + 2T^{2} \)
5 \( 1 + 3.03T + 5T^{2} \)
11 \( 1 - 6.14T + 11T^{2} \)
17 \( 1 - 6.44T + 17T^{2} \)
19 \( 1 - 4.32T + 19T^{2} \)
23 \( 1 + 8.92T + 23T^{2} \)
29 \( 1 + 2.44T + 29T^{2} \)
31 \( 1 - 3.00T + 31T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 + 0.945T + 41T^{2} \)
43 \( 1 + 5.28T + 43T^{2} \)
47 \( 1 + 0.212T + 47T^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 - 1.72T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 6.35T + 67T^{2} \)
71 \( 1 - 7.06T + 71T^{2} \)
73 \( 1 + 1.75T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 - 4.99T + 83T^{2} \)
89 \( 1 - 3.56T + 89T^{2} \)
97 \( 1 - 1.57T + 97T^{2} \)
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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

−8.411408952927072955278455818654, −7.903980498795821600533128739841, −7.56696734532001033016003972713, −6.56483743409865051513273433870, −5.44803554346066571775040824393, −4.43349078949349770355500932674, −3.83224606108091787498368238543, −3.39666760637699563683650262955, −1.66227577356014225012625103072, −0.791283921873866277957190995171, 0.791283921873866277957190995171, 1.66227577356014225012625103072, 3.39666760637699563683650262955, 3.83224606108091787498368238543, 4.43349078949349770355500932674, 5.44803554346066571775040824393, 6.56483743409865051513273433870, 7.56696734532001033016003972713, 7.903980498795821600533128739841, 8.411408952927072955278455818654

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