Properties

Degree 2
Conductor $ 2^{3} \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  + 0.763·3-s + 1.55·5-s − 7-s − 2.41·9-s + 11-s − 13-s + 1.18·15-s + 3.61·17-s + 1.02·19-s − 0.763·21-s + 4.55·23-s − 2.58·25-s − 4.13·27-s − 4.78·29-s + 2.84·31-s + 0.763·33-s − 1.55·35-s + 1.27·37-s − 0.763·39-s + 3.11·41-s + 8.35·43-s − 3.75·45-s + 9.30·47-s + 49-s + 2.75·51-s + 1.24·53-s + 1.55·55-s + ⋯
L(s)  = 1  + 0.440·3-s + 0.695·5-s − 0.377·7-s − 0.805·9-s + 0.301·11-s − 0.277·13-s + 0.306·15-s + 0.875·17-s + 0.235·19-s − 0.166·21-s + 0.949·23-s − 0.516·25-s − 0.796·27-s − 0.888·29-s + 0.511·31-s + 0.132·33-s − 0.262·35-s + 0.208·37-s − 0.122·39-s + 0.485·41-s + 1.27·43-s − 0.559·45-s + 1.35·47-s + 0.142·49-s + 0.386·51-s + 0.170·53-s + 0.209·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8008\)    =    \(2^{3} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.483556620$
$L(\frac12)$  $\approx$  $2.483556620$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 - 0.763T + 3T^{2} \)
5 \( 1 - 1.55T + 5T^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 - 1.02T + 19T^{2} \)
23 \( 1 - 4.55T + 23T^{2} \)
29 \( 1 + 4.78T + 29T^{2} \)
31 \( 1 - 2.84T + 31T^{2} \)
37 \( 1 - 1.27T + 37T^{2} \)
41 \( 1 - 3.11T + 41T^{2} \)
43 \( 1 - 8.35T + 43T^{2} \)
47 \( 1 - 9.30T + 47T^{2} \)
53 \( 1 - 1.24T + 53T^{2} \)
59 \( 1 + 2.95T + 59T^{2} \)
61 \( 1 + 6.44T + 61T^{2} \)
67 \( 1 + 6.51T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 6.95T + 73T^{2} \)
79 \( 1 - 2.23T + 79T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 - 1.06T + 97T^{2} \)
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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

−7.57693275097656366609957898250, −7.45567206344289384240845355621, −6.21011753653174823011559187444, −5.88255841326525140141645455618, −5.18192697035410484990325995166, −4.20647806019891590924206680710, −3.32766821915943797435071972090, −2.73279087154492495719502824886, −1.90556099116131800586494793652, −0.75783315802334772827118144531, 0.75783315802334772827118144531, 1.90556099116131800586494793652, 2.73279087154492495719502824886, 3.32766821915943797435071972090, 4.20647806019891590924206680710, 5.18192697035410484990325995166, 5.88255841326525140141645455618, 6.21011753653174823011559187444, 7.45567206344289384240845355621, 7.57693275097656366609957898250

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