Properties

Degree 2
Conductor $ 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.983·3-s − 1.20·5-s − 7-s − 2.03·9-s + 11-s − 13-s + 1.18·15-s + 3.10·17-s − 6.13·19-s + 0.983·21-s + 9.39·23-s − 3.54·25-s + 4.94·27-s − 2.82·29-s + 0.543·31-s − 0.983·33-s + 1.20·35-s − 0.252·37-s + 0.983·39-s + 0.665·41-s − 0.327·43-s + 2.45·45-s + 9.39·47-s + 49-s − 3.05·51-s + 8.35·53-s − 1.20·55-s + ⋯
L(s)  = 1  − 0.567·3-s − 0.539·5-s − 0.377·7-s − 0.677·9-s + 0.301·11-s − 0.277·13-s + 0.306·15-s + 0.753·17-s − 1.40·19-s + 0.214·21-s + 1.95·23-s − 0.708·25-s + 0.952·27-s − 0.523·29-s + 0.0975·31-s − 0.171·33-s + 0.204·35-s − 0.0415·37-s + 0.157·39-s + 0.103·41-s − 0.0498·43-s + 0.365·45-s + 1.37·47-s + 0.142·49-s − 0.427·51-s + 1.14·53-s − 0.162·55-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  =  1
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 0.983T + 3T^{2} \)
5 \( 1 + 1.20T + 5T^{2} \)
17 \( 1 - 3.10T + 17T^{2} \)
19 \( 1 + 6.13T + 19T^{2} \)
23 \( 1 - 9.39T + 23T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 - 0.543T + 31T^{2} \)
37 \( 1 + 0.252T + 37T^{2} \)
41 \( 1 - 0.665T + 41T^{2} \)
43 \( 1 + 0.327T + 43T^{2} \)
47 \( 1 - 9.39T + 47T^{2} \)
53 \( 1 - 8.35T + 53T^{2} \)
59 \( 1 - 8.49T + 59T^{2} \)
61 \( 1 + 4.81T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 9.69T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 + 9.44T + 83T^{2} \)
89 \( 1 + 3.56T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−7.34072822560880860834571819134, −6.85191681107406211474646527476, −6.01051197746719177540758007985, −5.52007294918048339676234966988, −4.68438496312592528860402042429, −3.93225990322162778153351192020, −3.14285298980434923660268966279, −2.33179910158826060240977524737, −1.00016009783383279374137696164, 0, 1.00016009783383279374137696164, 2.33179910158826060240977524737, 3.14285298980434923660268966279, 3.93225990322162778153351192020, 4.68438496312592528860402042429, 5.52007294918048339676234966988, 6.01051197746719177540758007985, 6.85191681107406211474646527476, 7.34072822560880860834571819134

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