Properties

Label 2-5733-1.1-c1-0-177
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  + 0.670·2-s − 1.54·4-s + 2.54·5-s − 2.38·8-s + 1.71·10-s + 3.05·11-s − 13-s + 1.50·16-s + 1.34·17-s − 7.60·19-s − 3.95·20-s + 2.04·22-s − 1.84·23-s + 1.50·25-s − 0.670·26-s + 5.60·29-s − 10.2·31-s + 5.77·32-s + 0.900·34-s − 9.20·37-s − 5.09·38-s − 6.07·40-s − 8.04·41-s + 1.49·43-s − 4.73·44-s − 1.23·46-s − 3.89·47-s + ⋯
L(s)  = 1  + 0.474·2-s − 0.774·4-s + 1.14·5-s − 0.841·8-s + 0.540·10-s + 0.920·11-s − 0.277·13-s + 0.375·16-s + 0.325·17-s − 1.74·19-s − 0.883·20-s + 0.436·22-s − 0.384·23-s + 0.300·25-s − 0.131·26-s + 1.04·29-s − 1.84·31-s + 1.02·32-s + 0.154·34-s − 1.51·37-s − 0.827·38-s − 0.960·40-s − 1.25·41-s + 0.228·43-s − 0.713·44-s − 0.182·46-s − 0.567·47-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: \(1\)
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 \( 1 - 0.670T + 2T^{2} \)
5 \( 1 - 2.54T + 5T^{2} \)
11 \( 1 - 3.05T + 11T^{2} \)
17 \( 1 - 1.34T + 17T^{2} \)
19 \( 1 + 7.60T + 19T^{2} \)
23 \( 1 + 1.84T + 23T^{2} \)
29 \( 1 - 5.60T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 9.20T + 37T^{2} \)
41 \( 1 + 8.04T + 41T^{2} \)
43 \( 1 - 1.49T + 43T^{2} \)
47 \( 1 + 3.89T + 47T^{2} \)
53 \( 1 + 0.502T + 53T^{2} \)
59 \( 1 + 4.28T + 59T^{2} \)
61 \( 1 - 0.683T + 61T^{2} \)
67 \( 1 + 7.68T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 12.0T + 73T^{2} \)
79 \( 1 - 4.91T + 79T^{2} \)
83 \( 1 + 1.20T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 7.18T + 97T^{2} \)
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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

−7.87735713388101808109938553190, −6.62088663607660779326135894315, −6.40380694864835354492132734570, −5.45917633279771461146456915261, −4.99353186951057381344112732701, −4.04982139901124565441565025778, −3.47090250326482962683171520744, −2.28927227031623131388669179107, −1.50751898291570420595350312586, 0, 1.50751898291570420595350312586, 2.28927227031623131388669179107, 3.47090250326482962683171520744, 4.04982139901124565441565025778, 4.99353186951057381344112732701, 5.45917633279771461146456915261, 6.40380694864835354492132734570, 6.62088663607660779326135894315, 7.87735713388101808109938553190

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