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  + 2.05·3-s + 0.929·5-s + 7-s + 1.22·9-s − 11-s − 13-s + 1.90·15-s − 5.05·17-s + 4.52·19-s + 2.05·21-s − 3.13·23-s − 4.13·25-s − 3.65·27-s − 3.89·29-s − 0.650·31-s − 2.05·33-s + 0.929·35-s − 6.58·37-s − 2.05·39-s − 11.3·41-s − 8.86·43-s + 1.13·45-s + 7.96·47-s + 49-s − 10.3·51-s − 9.49·53-s − 0.929·55-s + ⋯
L(s)  = 1  + 1.18·3-s + 0.415·5-s + 0.377·7-s + 0.407·9-s − 0.301·11-s − 0.277·13-s + 0.493·15-s − 1.22·17-s + 1.03·19-s + 0.448·21-s − 0.652·23-s − 0.827·25-s − 0.702·27-s − 0.722·29-s − 0.116·31-s − 0.357·33-s + 0.157·35-s − 1.08·37-s − 0.329·39-s − 1.77·41-s − 1.35·43-s + 0.169·45-s + 1.16·47-s + 0.142·49-s − 1.45·51-s − 1.30·53-s − 0.125·55-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 \( 1 - 2.05T + 3T^{2} \)
5 \( 1 - 0.929T + 5T^{2} \)
17 \( 1 + 5.05T + 17T^{2} \)
19 \( 1 - 4.52T + 19T^{2} \)
23 \( 1 + 3.13T + 23T^{2} \)
29 \( 1 + 3.89T + 29T^{2} \)
31 \( 1 + 0.650T + 31T^{2} \)
37 \( 1 + 6.58T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 8.86T + 43T^{2} \)
47 \( 1 - 7.96T + 47T^{2} \)
53 \( 1 + 9.49T + 53T^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 7.70T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 15.4T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 8.85T + 83T^{2} \)
89 \( 1 + 1.21T + 89T^{2} \)
97 \( 1 - 6.19T + 97T^{2} \)
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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

−7.58911096831359148284853253930, −7.01849454444486657849193441222, −6.09068919491276718870970230904, −5.34708726015915733266431434820, −4.64105723287671001278733406904, −3.69084319474283694058941396141, −3.11073145321539447667765909210, −2.11343078737310905842989164507, −1.74749695323256208775032025337, 0, 1.74749695323256208775032025337, 2.11343078737310905842989164507, 3.11073145321539447667765909210, 3.69084319474283694058941396141, 4.64105723287671001278733406904, 5.34708726015915733266431434820, 6.09068919491276718870970230904, 7.01849454444486657849193441222, 7.58911096831359148284853253930

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