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 + 2.59·3-s + 4-s + 5-s + 2.59·6-s + 1.31·7-s + 8-s + 3.75·9-s + 10-s − 11-s + 2.59·12-s − 5.57·13-s + 1.31·14-s + 2.59·15-s + 16-s + 6.71·17-s + 3.75·18-s − 4.17·19-s + 20-s + 3.41·21-s − 22-s + 7.57·23-s + 2.59·24-s + 25-s − 5.57·26-s + 1.96·27-s + 1.31·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.50·3-s + 0.5·4-s + 0.447·5-s + 1.06·6-s + 0.496·7-s + 0.353·8-s + 1.25·9-s + 0.316·10-s − 0.301·11-s + 0.750·12-s − 1.54·13-s + 0.351·14-s + 0.671·15-s + 0.250·16-s + 1.62·17-s + 0.885·18-s − 0.958·19-s + 0.223·20-s + 0.745·21-s − 0.213·22-s + 1.57·23-s + 0.530·24-s + 0.200·25-s − 1.09·26-s + 0.379·27-s + 0.248·28-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  =  $(2,\ 4730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.209840328$
$L(\frac12)$  $\approx$  $6.209840328$
$L(\frac{3}{2})$   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 \( 1 - 2.59T + 3T^{2} \)
7 \( 1 - 1.31T + 7T^{2} \)
13 \( 1 + 5.57T + 13T^{2} \)
17 \( 1 - 6.71T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 7.57T + 23T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 - 5.06T + 31T^{2} \)
37 \( 1 + 4.21T + 37T^{2} \)
41 \( 1 - 8.18T + 41T^{2} \)
47 \( 1 - 9.44T + 47T^{2} \)
53 \( 1 + 7.39T + 53T^{2} \)
59 \( 1 + 9.36T + 59T^{2} \)
61 \( 1 + 0.668T + 61T^{2} \)
67 \( 1 + 2.23T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 - 5.06T + 73T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 6.57T + 97T^{2} \)
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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

−8.034725882113262284080961894037, −7.73616556217183655404543906306, −7.00919148968145542133324616572, −6.07929231557880763636946797454, −5.05112110693599338358887386898, −4.67030628072510775528319600189, −3.57913394177912773525807725477, −2.77763571056301780678254657798, −2.37896428838326776495196291362, −1.27765851446567047280402261180, 1.27765851446567047280402261180, 2.37896428838326776495196291362, 2.77763571056301780678254657798, 3.57913394177912773525807725477, 4.67030628072510775528319600189, 5.05112110693599338358887386898, 6.07929231557880763636946797454, 7.00919148968145542133324616572, 7.73616556217183655404543906306, 8.034725882113262284080961894037

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