Properties

Label 2-5733-1.1-c1-0-119
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44·2-s + 3.99·4-s + 0.910·5-s + 4.87·8-s + 2.22·10-s − 3.67·11-s − 13-s + 3.95·16-s + 7.18·17-s + 1.97·19-s + 3.63·20-s − 9.00·22-s + 0.596·23-s − 4.17·25-s − 2.44·26-s + 3.64·29-s + 7.08·31-s − 0.0786·32-s + 17.5·34-s + 0.710·37-s + 4.84·38-s + 4.43·40-s + 5.27·41-s + 11.0·43-s − 14.6·44-s + 1.46·46-s + 12.1·47-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.99·4-s + 0.407·5-s + 1.72·8-s + 0.704·10-s − 1.10·11-s − 0.277·13-s + 0.987·16-s + 1.74·17-s + 0.453·19-s + 0.812·20-s − 1.91·22-s + 0.124·23-s − 0.834·25-s − 0.480·26-s + 0.677·29-s + 1.27·31-s − 0.0139·32-s + 3.01·34-s + 0.116·37-s + 0.785·38-s + 0.701·40-s + 0.823·41-s + 1.68·43-s − 2.21·44-s + 0.215·46-s + 1.76·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5733,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.510829253\)
\(L(\frac12)\) \(\approx\) \(6.510829253\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.44T + 2T^{2} \)
5 \( 1 - 0.910T + 5T^{2} \)
11 \( 1 + 3.67T + 11T^{2} \)
17 \( 1 - 7.18T + 17T^{2} \)
19 \( 1 - 1.97T + 19T^{2} \)
23 \( 1 - 0.596T + 23T^{2} \)
29 \( 1 - 3.64T + 29T^{2} \)
31 \( 1 - 7.08T + 31T^{2} \)
37 \( 1 - 0.710T + 37T^{2} \)
41 \( 1 - 5.27T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 9.58T + 59T^{2} \)
61 \( 1 + 6.98T + 61T^{2} \)
67 \( 1 - 1.22T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 6.53T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 7.16T + 83T^{2} \)
89 \( 1 - 12.8T + 89T^{2} \)
97 \( 1 + 9.09T + 97T^{2} \)
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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

−7.55672720972543977570327098006, −7.50611141007961290864308855772, −6.33199337022978923502086042326, −5.61763954418002696913496740009, −5.43029422686498172875242121096, −4.48267677594962619251184766592, −3.80712864267146896367255347370, −2.76088185299542786441067534025, −2.51122998091485589399661627935, −1.08979065815532064532752235635, 1.08979065815532064532752235635, 2.51122998091485589399661627935, 2.76088185299542786441067534025, 3.80712864267146896367255347370, 4.48267677594962619251184766592, 5.43029422686498172875242121096, 5.61763954418002696913496740009, 6.33199337022978923502086042326, 7.50611141007961290864308855772, 7.55672720972543977570327098006

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