Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 11 \cdot 43 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.80·3-s + 4-s − 5-s − 1.80·6-s + 4.53·7-s − 8-s + 0.273·9-s + 10-s − 11-s + 1.80·12-s + 1.28·13-s − 4.53·14-s − 1.80·15-s + 16-s − 2.49·17-s − 0.273·18-s + 4.30·19-s − 20-s + 8.21·21-s + 22-s − 0.713·23-s − 1.80·24-s + 25-s − 1.28·26-s − 4.93·27-s + 4.53·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.04·3-s + 0.5·4-s − 0.447·5-s − 0.738·6-s + 1.71·7-s − 0.353·8-s + 0.0912·9-s + 0.316·10-s − 0.301·11-s + 0.522·12-s + 0.356·13-s − 1.21·14-s − 0.467·15-s + 0.250·16-s − 0.606·17-s − 0.0644·18-s + 0.987·19-s − 0.223·20-s + 1.79·21-s + 0.213·22-s − 0.148·23-s − 0.369·24-s + 0.200·25-s − 0.252·26-s − 0.949·27-s + 0.857·28-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

\( d \)  =  \(2\)
\( N \)  =  \(4730\)    =    \(2 \cdot 5 \cdot 11 \cdot 43\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4730} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.310279263$
$L(\frac12)$  $\approx$  $2.310279263$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;43\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 + T \)
43 \( 1 - T \)
good3 \( 1 - 1.80T + 3T^{2} \)
7 \( 1 - 4.53T + 7T^{2} \)
13 \( 1 - 1.28T + 13T^{2} \)
17 \( 1 + 2.49T + 17T^{2} \)
19 \( 1 - 4.30T + 19T^{2} \)
23 \( 1 + 0.713T + 23T^{2} \)
29 \( 1 + 4.09T + 29T^{2} \)
31 \( 1 - 6.87T + 31T^{2} \)
37 \( 1 - 2.58T + 37T^{2} \)
41 \( 1 - 2.61T + 41T^{2} \)
47 \( 1 - 7.75T + 47T^{2} \)
53 \( 1 - 7.15T + 53T^{2} \)
59 \( 1 + 2.05T + 59T^{2} \)
61 \( 1 + 7.20T + 61T^{2} \)
67 \( 1 - 4.88T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 4.33T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 - 3.85T + 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.292953436851440439080476222201, −7.62835152393624973167921883625, −7.50385190795608767834404628488, −6.21401324603192724851426544432, −5.31100348402051851886551767365, −4.48636818453412205789187051538, −3.63707110136755055585157456020, −2.64605146804612419610100810680, −1.96356750447216182688987730020, −0.919206797848796634323582972701, 0.919206797848796634323582972701, 1.96356750447216182688987730020, 2.64605146804612419610100810680, 3.63707110136755055585157456020, 4.48636818453412205789187051538, 5.31100348402051851886551767365, 6.21401324603192724851426544432, 7.50385190795608767834404628488, 7.62835152393624973167921883625, 8.292953436851440439080476222201

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