Properties

Label 2-5733-1.1-c1-0-139
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  − 0.470·2-s − 1.77·4-s + 0.529·5-s + 1.77·8-s − 0.249·10-s + 2.24·11-s − 13-s + 2.71·16-s − 1.30·17-s + 1.47·19-s − 0.941·20-s − 1.05·22-s − 5.83·23-s − 4.71·25-s + 0.470·26-s − 5.22·29-s + 7.02·31-s − 4.83·32-s + 0.615·34-s − 2.36·37-s − 0.692·38-s + 0.941·40-s + 6.49·41-s + 11.3·43-s − 4.00·44-s + 2.74·46-s + 8.58·47-s + ⋯
L(s)  = 1  − 0.332·2-s − 0.889·4-s + 0.236·5-s + 0.628·8-s − 0.0787·10-s + 0.678·11-s − 0.277·13-s + 0.679·16-s − 0.317·17-s + 0.337·19-s − 0.210·20-s − 0.225·22-s − 1.21·23-s − 0.943·25-s + 0.0923·26-s − 0.969·29-s + 1.26·31-s − 0.855·32-s + 0.105·34-s − 0.389·37-s − 0.112·38-s + 0.148·40-s + 1.01·41-s + 1.73·43-s − 0.603·44-s + 0.405·46-s + 1.25·47-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 \( 1 + 0.470T + 2T^{2} \)
5 \( 1 - 0.529T + 5T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
17 \( 1 + 1.30T + 17T^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 + 5.83T + 23T^{2} \)
29 \( 1 + 5.22T + 29T^{2} \)
31 \( 1 - 7.02T + 31T^{2} \)
37 \( 1 + 2.36T + 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 - 11.3T + 43T^{2} \)
47 \( 1 - 8.58T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 + 1.19T + 71T^{2} \)
73 \( 1 + 7.64T + 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 - 3.47T + 97T^{2} \)
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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

−7.66325094295565002468327797602, −7.41490125714622245671848529832, −6.00173115838023391012777872041, −5.90859225246227753217270002182, −4.61848616347178031920120262817, −4.25383830594107742839147924885, −3.35243017741884140470896179548, −2.18648840239592804521063851983, −1.21638248487755075255628957140, 0, 1.21638248487755075255628957140, 2.18648840239592804521063851983, 3.35243017741884140470896179548, 4.25383830594107742839147924885, 4.61848616347178031920120262817, 5.90859225246227753217270002182, 6.00173115838023391012777872041, 7.41490125714622245671848529832, 7.66325094295565002468327797602

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