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  − 1.55·3-s − 0.910·5-s − 7-s − 0.592·9-s − 11-s − 13-s + 1.41·15-s + 6.01·17-s − 0.663·19-s + 1.55·21-s + 5.85·23-s − 4.17·25-s + 5.57·27-s − 3.16·29-s − 9.74·31-s + 1.55·33-s + 0.910·35-s − 2.95·37-s + 1.55·39-s − 3.08·41-s + 5.25·43-s + 0.539·45-s + 6.11·47-s + 49-s − 9.33·51-s + 9.32·53-s + 0.910·55-s + ⋯
L(s)  = 1  − 0.895·3-s − 0.407·5-s − 0.377·7-s − 0.197·9-s − 0.301·11-s − 0.277·13-s + 0.364·15-s + 1.45·17-s − 0.152·19-s + 0.338·21-s + 1.22·23-s − 0.834·25-s + 1.07·27-s − 0.586·29-s − 1.74·31-s + 0.270·33-s + 0.153·35-s − 0.485·37-s + 0.248·39-s − 0.481·41-s + 0.802·43-s + 0.0803·45-s + 0.892·47-s + 0.142·49-s − 1.30·51-s + 1.28·53-s + 0.122·55-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 \( 1 + 1.55T + 3T^{2} \)
5 \( 1 + 0.910T + 5T^{2} \)
17 \( 1 - 6.01T + 17T^{2} \)
19 \( 1 + 0.663T + 19T^{2} \)
23 \( 1 - 5.85T + 23T^{2} \)
29 \( 1 + 3.16T + 29T^{2} \)
31 \( 1 + 9.74T + 31T^{2} \)
37 \( 1 + 2.95T + 37T^{2} \)
41 \( 1 + 3.08T + 41T^{2} \)
43 \( 1 - 5.25T + 43T^{2} \)
47 \( 1 - 6.11T + 47T^{2} \)
53 \( 1 - 9.32T + 53T^{2} \)
59 \( 1 + 5.66T + 59T^{2} \)
61 \( 1 - 0.357T + 61T^{2} \)
67 \( 1 - 6.56T + 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + 3.71T + 73T^{2} \)
79 \( 1 - 1.33T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 - 0.721T + 89T^{2} \)
97 \( 1 - 4.59T + 97T^{2} \)
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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

−7.39933686569091551993935142899, −6.86563692046361293934303334983, −5.86960756139028164265481290598, −5.51792274445486872192013960863, −4.89401448549340882173763539952, −3.81524355525294658388608278745, −3.26200035161576270028034101515, −2.23653456983946074040187151589, −0.968200407501423047824460775005, 0, 0.968200407501423047824460775005, 2.23653456983946074040187151589, 3.26200035161576270028034101515, 3.81524355525294658388608278745, 4.89401448549340882173763539952, 5.51792274445486872192013960863, 5.86960756139028164265481290598, 6.86563692046361293934303334983, 7.39933686569091551993935142899

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