Properties

Label 2-4730-1.1-c1-0-62
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.86·3-s + 4-s + 5-s + 2.86·6-s − 1.78·7-s − 8-s + 5.19·9-s − 10-s + 11-s − 2.86·12-s − 5.69·13-s + 1.78·14-s − 2.86·15-s + 16-s + 0.963·17-s − 5.19·18-s + 4.63·19-s + 20-s + 5.12·21-s − 22-s − 7.69·23-s + 2.86·24-s + 25-s + 5.69·26-s − 6.28·27-s − 1.78·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.65·3-s + 0.5·4-s + 0.447·5-s + 1.16·6-s − 0.676·7-s − 0.353·8-s + 1.73·9-s − 0.316·10-s + 0.301·11-s − 0.826·12-s − 1.58·13-s + 0.478·14-s − 0.739·15-s + 0.250·16-s + 0.233·17-s − 1.22·18-s + 1.06·19-s + 0.223·20-s + 1.11·21-s − 0.213·22-s − 1.60·23-s + 0.584·24-s + 0.200·25-s + 1.11·26-s − 1.20·27-s − 0.338·28-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: \(1\)
Selberg data: \((2,\ 4730,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 - T \)
43 \( 1 + T \)
good3 \( 1 + 2.86T + 3T^{2} \)
7 \( 1 + 1.78T + 7T^{2} \)
13 \( 1 + 5.69T + 13T^{2} \)
17 \( 1 - 0.963T + 17T^{2} \)
19 \( 1 - 4.63T + 19T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 - 4.30T + 37T^{2} \)
41 \( 1 + 4.75T + 41T^{2} \)
47 \( 1 - 7.87T + 47T^{2} \)
53 \( 1 + 3.39T + 53T^{2} \)
59 \( 1 + 6.29T + 59T^{2} \)
61 \( 1 - 3.05T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 + 6.14T + 79T^{2} \)
83 \( 1 - 2.53T + 83T^{2} \)
89 \( 1 + 3.42T + 89T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
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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

−7.66329545464333117770565097766, −7.17242372551812700355423994899, −6.34843348904324092267047891254, −5.98590093083542838492242965937, −5.13713378386452016179908879800, −4.49087389034731714289884729078, −3.20854368031342544361032400673, −2.13838263193937889778502939675, −0.978337269785765959093172738511, 0, 0.978337269785765959093172738511, 2.13838263193937889778502939675, 3.20854368031342544361032400673, 4.49087389034731714289884729078, 5.13713378386452016179908879800, 5.98590093083542838492242965937, 6.34843348904324092267047891254, 7.17242372551812700355423994899, 7.66329545464333117770565097766

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