Properties

Label 2-5733-1.1-c1-0-110
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  + 2.61·2-s + 4.85·4-s − 2.23·5-s + 7.47·8-s − 5.85·10-s + 3·11-s − 13-s + 9.85·16-s − 1.47·17-s + 3·19-s − 10.8·20-s + 7.85·22-s + 8.23·23-s − 2.61·26-s − 4.47·29-s + 5·31-s + 10.8·32-s − 3.85·34-s + 4.70·37-s + 7.85·38-s − 16.7·40-s + 4.47·41-s − 8·43-s + 14.5·44-s + 21.5·46-s + 7.47·47-s − 4.85·52-s + ⋯
L(s)  = 1  + 1.85·2-s + 2.42·4-s − 0.999·5-s + 2.64·8-s − 1.85·10-s + 0.904·11-s − 0.277·13-s + 2.46·16-s − 0.357·17-s + 0.688·19-s − 2.42·20-s + 1.67·22-s + 1.71·23-s − 0.513·26-s − 0.830·29-s + 0.898·31-s + 1.91·32-s − 0.660·34-s + 0.774·37-s + 1.27·38-s − 2.64·40-s + 0.698·41-s − 1.21·43-s + 2.19·44-s + 3.17·46-s + 1.08·47-s − 0.673·52-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: \(0\)
Selberg data: \((2,\ 5733,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.240507242\)
\(L(\frac12)\) \(\approx\) \(6.240507242\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 2T^{2} \)
5 \( 1 + 2.23T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 - 8.23T + 23T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - 4.70T + 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 7.47T + 47T^{2} \)
53 \( 1 - 7.47T + 53T^{2} \)
59 \( 1 - 1.47T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 - 8.94T + 71T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 + 2.70T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 - 9.41T + 97T^{2} \)
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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

−7.69629757132774707611651376452, −7.17955611398549923853908406109, −6.62305805131218474681205173489, −5.81398570354841287006827766299, −5.04922240682503243801843755772, −4.40802129489538029928248668667, −3.79572697239028377233237807751, −3.16810263603139382652102846667, −2.31518502414556948623382678579, −1.05093587368298752338815604664, 1.05093587368298752338815604664, 2.31518502414556948623382678579, 3.16810263603139382652102846667, 3.79572697239028377233237807751, 4.40802129489538029928248668667, 5.04922240682503243801843755772, 5.81398570354841287006827766299, 6.62305805131218474681205173489, 7.17955611398549923853908406109, 7.69629757132774707611651376452

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