Properties

Label 2-2013-1.1-c1-0-39
Degree $2$
Conductor $2013$
Sign $-1$
Analytic cond. $16.0738$
Root an. cond. $4.00922$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·2-s + 3-s + 5.47·4-s − 3.64·5-s − 2.73·6-s − 3.96·7-s − 9.49·8-s + 9-s + 9.96·10-s + 11-s + 5.47·12-s + 2.49·13-s + 10.8·14-s − 3.64·15-s + 14.9·16-s − 4.84·17-s − 2.73·18-s − 0.0465·19-s − 19.9·20-s − 3.96·21-s − 2.73·22-s + 7.87·23-s − 9.49·24-s + 8.30·25-s − 6.81·26-s + 27-s − 21.6·28-s + ⋯
L(s)  = 1  − 1.93·2-s + 0.577·3-s + 2.73·4-s − 1.63·5-s − 1.11·6-s − 1.49·7-s − 3.35·8-s + 0.333·9-s + 3.15·10-s + 0.301·11-s + 1.57·12-s + 0.691·13-s + 2.89·14-s − 0.941·15-s + 3.74·16-s − 1.17·17-s − 0.644·18-s − 0.0106·19-s − 4.46·20-s − 0.864·21-s − 0.582·22-s + 1.64·23-s − 1.93·24-s + 1.66·25-s − 1.33·26-s + 0.192·27-s − 4.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(16.0738\)
Root analytic conductor: \(4.00922\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2013,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 - T \)
61 \( 1 + T \)
good2 \( 1 + 2.73T + 2T^{2} \)
5 \( 1 + 3.64T + 5T^{2} \)
7 \( 1 + 3.96T + 7T^{2} \)
13 \( 1 - 2.49T + 13T^{2} \)
17 \( 1 + 4.84T + 17T^{2} \)
19 \( 1 + 0.0465T + 19T^{2} \)
23 \( 1 - 7.87T + 23T^{2} \)
29 \( 1 - 4.43T + 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 + 1.28T + 41T^{2} \)
43 \( 1 - 6.56T + 43T^{2} \)
47 \( 1 - 0.000570T + 47T^{2} \)
53 \( 1 + 9.54T + 53T^{2} \)
59 \( 1 + 7.63T + 59T^{2} \)
67 \( 1 + 8.92T + 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 8.55T + 79T^{2} \)
83 \( 1 - 3.71T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 6.11T + 97T^{2} \)
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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

−8.750520696770591140398285401526, −8.263654375972811654656358146105, −7.37222564282430485572232515333, −6.78599327827427905738209226489, −6.30648875761418726939987016509, −4.36155864319286443386223699656, −3.21577028503659093326318284442, −2.82149408645607506802514439416, −1.10574246248730747778597475428, 0, 1.10574246248730747778597475428, 2.82149408645607506802514439416, 3.21577028503659093326318284442, 4.36155864319286443386223699656, 6.30648875761418726939987016509, 6.78599327827427905738209226489, 7.37222564282430485572232515333, 8.263654375972811654656358146105, 8.750520696770591140398285401526

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