Properties

Label 2-5733-1.1-c1-0-200
Degree $2$
Conductor $5733$
Sign $-1$
Analytic cond. $45.7782$
Root an. cond. $6.76596$
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  + 1.81·2-s + 1.28·4-s + 2.81·5-s − 1.28·8-s + 5.10·10-s − 3.10·11-s − 13-s − 4.91·16-s − 0.524·17-s − 0.813·19-s + 3.62·20-s − 5.62·22-s − 7.33·23-s + 2.91·25-s − 1.81·26-s − 8.28·29-s − 1.39·31-s − 6.33·32-s − 0.951·34-s − 6.15·37-s − 1.47·38-s − 3.62·40-s − 4.20·41-s + 6.75·43-s − 3.99·44-s − 13.3·46-s − 5.97·47-s + ⋯
L(s)  = 1  + 1.28·2-s + 0.644·4-s + 1.25·5-s − 0.455·8-s + 1.61·10-s − 0.935·11-s − 0.277·13-s − 1.22·16-s − 0.127·17-s − 0.186·19-s + 0.811·20-s − 1.19·22-s − 1.53·23-s + 0.583·25-s − 0.355·26-s − 1.53·29-s − 0.250·31-s − 1.12·32-s − 0.163·34-s − 1.01·37-s − 0.239·38-s − 0.573·40-s − 0.656·41-s + 1.03·43-s − 0.603·44-s − 1.96·46-s − 0.870·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5733\)    =    \(3^{2} \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(45.7782\)
Root analytic conductor: \(6.76596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5733} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5733,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
13 \( 1 + T \)
good2 \( 1 - 1.81T + 2T^{2} \)
5 \( 1 - 2.81T + 5T^{2} \)
11 \( 1 + 3.10T + 11T^{2} \)
17 \( 1 + 0.524T + 17T^{2} \)
19 \( 1 + 0.813T + 19T^{2} \)
23 \( 1 + 7.33T + 23T^{2} \)
29 \( 1 + 8.28T + 29T^{2} \)
31 \( 1 + 1.39T + 31T^{2} \)
37 \( 1 + 6.15T + 37T^{2} \)
41 \( 1 + 4.20T + 41T^{2} \)
43 \( 1 - 6.75T + 43T^{2} \)
47 \( 1 + 5.97T + 47T^{2} \)
53 \( 1 - 2.49T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 8.72T + 71T^{2} \)
73 \( 1 - 2.34T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 16.4T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 1.18T + 97T^{2} \)
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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

−7.61929158389835206292358108821, −6.73557186223943787482678231608, −6.04506516239010667841791313432, −5.49495758383364889825744542396, −5.06898064119564441732956838655, −4.11710879585163276589862533402, −3.37365015269456516225678205651, −2.35769717024745362005061856055, −1.91024563837143301616346394150, 0, 1.91024563837143301616346394150, 2.35769717024745362005061856055, 3.37365015269456516225678205651, 4.11710879585163276589862533402, 5.06898064119564441732956838655, 5.49495758383364889825744542396, 6.04506516239010667841791313432, 6.73557186223943787482678231608, 7.61929158389835206292358108821

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