Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.497·3-s − 2.93·5-s − 7-s − 2.75·9-s − 11-s − 13-s − 1.46·15-s + 6.64·17-s − 8.43·19-s − 0.497·21-s − 7.56·23-s + 3.64·25-s − 2.85·27-s − 0.0222·29-s + 1.14·31-s − 0.497·33-s + 2.93·35-s − 0.840·37-s − 0.497·39-s + 4.48·41-s + 1.41·43-s + 8.09·45-s + 6.97·47-s + 49-s + 3.30·51-s + 11.8·53-s + 2.93·55-s + ⋯
L(s)  = 1  + 0.286·3-s − 1.31·5-s − 0.377·7-s − 0.917·9-s − 0.301·11-s − 0.277·13-s − 0.377·15-s + 1.61·17-s − 1.93·19-s − 0.108·21-s − 1.57·23-s + 0.728·25-s − 0.550·27-s − 0.00413·29-s + 0.205·31-s − 0.0865·33-s + 0.496·35-s − 0.138·37-s − 0.0795·39-s + 0.699·41-s + 0.215·43-s + 1.20·45-s + 1.01·47-s + 0.142·49-s + 0.462·51-s + 1.62·53-s + 0.396·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7685878615$
$L(\frac12)$  $\approx$  $0.7685878615$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 - 0.497T + 3T^{2} \)
5 \( 1 + 2.93T + 5T^{2} \)
17 \( 1 - 6.64T + 17T^{2} \)
19 \( 1 + 8.43T + 19T^{2} \)
23 \( 1 + 7.56T + 23T^{2} \)
29 \( 1 + 0.0222T + 29T^{2} \)
31 \( 1 - 1.14T + 31T^{2} \)
37 \( 1 + 0.840T + 37T^{2} \)
41 \( 1 - 4.48T + 41T^{2} \)
43 \( 1 - 1.41T + 43T^{2} \)
47 \( 1 - 6.97T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 0.878T + 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 - 8.34T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 8.97T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 2.63T + 83T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 - 19.2T + 97T^{2} \)
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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

−8.433656477746637110598916967460, −7.75533903155029390904496114063, −7.26554360766423895616793149829, −6.10827369423426714848875762258, −5.63891718744136067973400285480, −4.39965026836652599064132745467, −3.86064062831267927307001637224, −3.04836586580854674071959494159, −2.17570798896445772719305871812, −0.46658448983115109548011475678, 0.46658448983115109548011475678, 2.17570798896445772719305871812, 3.04836586580854674071959494159, 3.86064062831267927307001637224, 4.39965026836652599064132745467, 5.63891718744136067973400285480, 6.10827369423426714848875762258, 7.26554360766423895616793149829, 7.75533903155029390904496114063, 8.433656477746637110598916967460

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