Properties

Label 2-5733-1.1-c1-0-80
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  + 0.381·2-s − 1.85·4-s + 2.23·5-s − 1.47·8-s + 0.854·10-s + 3·11-s − 13-s + 3.14·16-s + 7.47·17-s + 3·19-s − 4.14·20-s + 1.14·22-s + 3.76·23-s − 0.381·26-s + 4.47·29-s + 5·31-s + 4.14·32-s + 2.85·34-s − 8.70·37-s + 1.14·38-s − 3.29·40-s − 4.47·41-s − 8·43-s − 5.56·44-s + 1.43·46-s − 1.47·47-s + 1.85·52-s + ⋯
L(s)  = 1  + 0.270·2-s − 0.927·4-s + 0.999·5-s − 0.520·8-s + 0.270·10-s + 0.904·11-s − 0.277·13-s + 0.786·16-s + 1.81·17-s + 0.688·19-s − 0.927·20-s + 0.244·22-s + 0.784·23-s − 0.0749·26-s + 0.830·29-s + 0.898·31-s + 0.732·32-s + 0.489·34-s − 1.43·37-s + 0.185·38-s − 0.520·40-s − 0.698·41-s − 1.21·43-s − 0.838·44-s + 0.211·46-s − 0.214·47-s + 0.257·52-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

\(L(1)\) \(\approx\) \(2.539561907\)
\(L(\frac12)\) \(\approx\) \(2.539561907\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.381T + 2T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
17 \( 1 - 7.47T + 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 - 3.76T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + 8.70T + 37T^{2} \)
41 \( 1 + 4.47T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 1.47T + 47T^{2} \)
53 \( 1 + 1.47T + 53T^{2} \)
59 \( 1 + 7.47T + 59T^{2} \)
61 \( 1 - 3T + 61T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 2.23T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−8.270612109086063232287708959359, −7.39496068555586426443115287178, −6.52839518284144799122193282594, −5.86322008664896671218046157369, −5.17501842036711965255703147720, −4.69777243122186607431825797567, −3.50998178364644846807811630612, −3.11108144127331549652676534880, −1.70417661669501108230058458565, −0.883430952743246814743964384621, 0.883430952743246814743964384621, 1.70417661669501108230058458565, 3.11108144127331549652676534880, 3.50998178364644846807811630612, 4.69777243122186607431825797567, 5.17501842036711965255703147720, 5.86322008664896671218046157369, 6.52839518284144799122193282594, 7.39496068555586426443115287178, 8.270612109086063232287708959359

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