Properties

Label 2-5733-1.1-c1-0-112
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  − 2.34·2-s + 3.48·4-s − 1.34·5-s − 3.48·8-s + 3.14·10-s − 1.14·11-s − 13-s + 1.19·16-s + 5.83·17-s + 3.34·19-s − 4.68·20-s + 2.68·22-s + 3.17·23-s − 3.19·25-s + 2.34·26-s − 10.4·29-s − 1.63·31-s + 4.17·32-s − 13.6·34-s + 8.51·37-s − 7.83·38-s + 4.68·40-s − 0.292·41-s − 8.15·43-s − 4.00·44-s − 7.43·46-s − 10.6·47-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.74·4-s − 0.600·5-s − 1.23·8-s + 0.994·10-s − 0.345·11-s − 0.277·13-s + 0.299·16-s + 1.41·17-s + 0.766·19-s − 1.04·20-s + 0.572·22-s + 0.662·23-s − 0.639·25-s + 0.459·26-s − 1.94·29-s − 0.293·31-s + 0.738·32-s − 2.34·34-s + 1.40·37-s − 1.27·38-s + 0.740·40-s − 0.0457·41-s − 1.24·43-s − 0.603·44-s − 1.09·46-s − 1.54·47-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 \( 1 + 2.34T + 2T^{2} \)
5 \( 1 + 1.34T + 5T^{2} \)
11 \( 1 + 1.14T + 11T^{2} \)
17 \( 1 - 5.83T + 17T^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 - 3.17T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + 1.63T + 31T^{2} \)
37 \( 1 - 8.51T + 37T^{2} \)
41 \( 1 + 0.292T + 41T^{2} \)
43 \( 1 + 8.15T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 0.782T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 6.10T + 67T^{2} \)
71 \( 1 + 1.53T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 0.882T + 79T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 - 5.73T + 89T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
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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

−7.82430375515476795191906851658, −7.45301200944173434401777577436, −6.74806846502437154940726323465, −5.72927753631409768565412057026, −5.03536941510938430234784295977, −3.81766842303496704797449335203, −3.06182567836368138698080706012, −2.00141128250161081989778808507, −1.05568239992044774063077333034, 0, 1.05568239992044774063077333034, 2.00141128250161081989778808507, 3.06182567836368138698080706012, 3.81766842303496704797449335203, 5.03536941510938430234784295977, 5.72927753631409768565412057026, 6.74806846502437154940726323465, 7.45301200944173434401777577436, 7.82430375515476795191906851658

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