Properties

Label 2-4730-1.1-c1-0-47
Degree $2$
Conductor $4730$
Sign $1$
Analytic cond. $37.7692$
Root an. cond. $6.14566$
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-s − 1.33·3-s + 4-s + 5-s − 1.33·6-s − 0.356·7-s + 8-s − 1.20·9-s + 10-s + 11-s − 1.33·12-s + 3.42·13-s − 0.356·14-s − 1.33·15-s + 16-s + 4.44·17-s − 1.20·18-s − 0.263·19-s + 20-s + 0.477·21-s + 22-s − 1.63·23-s − 1.33·24-s + 25-s + 3.42·26-s + 5.63·27-s − 0.356·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.773·3-s + 0.5·4-s + 0.447·5-s − 0.546·6-s − 0.134·7-s + 0.353·8-s − 0.402·9-s + 0.316·10-s + 0.301·11-s − 0.386·12-s + 0.950·13-s − 0.0952·14-s − 0.345·15-s + 0.250·16-s + 1.07·17-s − 0.284·18-s − 0.0605·19-s + 0.223·20-s + 0.104·21-s + 0.213·22-s − 0.341·23-s − 0.273·24-s + 0.200·25-s + 0.671·26-s + 1.08·27-s − 0.0673·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
Sign: $1$
Analytic conductor: \(37.7692\)
Root analytic conductor: \(6.14566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4730,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.613404378\)
\(L(\frac12)\) \(\approx\) \(2.613404378\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 + T \)
good3 \( 1 + 1.33T + 3T^{2} \)
7 \( 1 + 0.356T + 7T^{2} \)
13 \( 1 - 3.42T + 13T^{2} \)
17 \( 1 - 4.44T + 17T^{2} \)
19 \( 1 + 0.263T + 19T^{2} \)
23 \( 1 + 1.63T + 23T^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 - 3.17T + 31T^{2} \)
37 \( 1 + 10.3T + 37T^{2} \)
41 \( 1 - 7.63T + 41T^{2} \)
47 \( 1 - 6.20T + 47T^{2} \)
53 \( 1 + 5.75T + 53T^{2} \)
59 \( 1 + 6.11T + 59T^{2} \)
61 \( 1 - 13.3T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + 0.541T + 71T^{2} \)
73 \( 1 - 11.2T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 5.07T + 89T^{2} \)
97 \( 1 - 6.53T + 97T^{2} \)
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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

−8.235266775347256813902193946269, −7.40246391650763433397591560909, −6.43132275642885151380395544761, −6.10343694852543194695430987949, −5.43134986841003251702600495794, −4.78476164805652228176122723921, −3.73075181279707753785346123755, −3.09812330582270080311692621794, −1.93520104703070196093022463452, −0.857121845332315242430450258225, 0.857121845332315242430450258225, 1.93520104703070196093022463452, 3.09812330582270080311692621794, 3.73075181279707753785346123755, 4.78476164805652228176122723921, 5.43134986841003251702600495794, 6.10343694852543194695430987949, 6.43132275642885151380395544761, 7.40246391650763433397591560909, 8.235266775347256813902193946269

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