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  − 1.48·3-s − 0.395·5-s − 7-s − 0.792·9-s + 11-s − 13-s + 0.587·15-s − 2.74·17-s + 5.24·19-s + 1.48·21-s − 2.01·23-s − 4.84·25-s + 5.63·27-s + 5.19·29-s − 8.17·31-s − 1.48·33-s + 0.395·35-s + 3.28·37-s + 1.48·39-s + 9.42·41-s + 6.55·43-s + 0.313·45-s − 2.01·47-s + 49-s + 4.08·51-s − 4.08·53-s − 0.395·55-s + ⋯
L(s)  = 1  − 0.857·3-s − 0.176·5-s − 0.377·7-s − 0.264·9-s + 0.301·11-s − 0.277·13-s + 0.151·15-s − 0.666·17-s + 1.20·19-s + 0.324·21-s − 0.420·23-s − 0.968·25-s + 1.08·27-s + 0.965·29-s − 1.46·31-s − 0.258·33-s + 0.0668·35-s + 0.540·37-s + 0.237·39-s + 1.47·41-s + 1.00·43-s + 0.0467·45-s − 0.294·47-s + 0.142·49-s + 0.571·51-s − 0.561·53-s − 0.0532·55-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 \( 1 + 1.48T + 3T^{2} \)
5 \( 1 + 0.395T + 5T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 - 5.24T + 19T^{2} \)
23 \( 1 + 2.01T + 23T^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 8.17T + 31T^{2} \)
37 \( 1 - 3.28T + 37T^{2} \)
41 \( 1 - 9.42T + 41T^{2} \)
43 \( 1 - 6.55T + 43T^{2} \)
47 \( 1 + 2.01T + 47T^{2} \)
53 \( 1 + 4.08T + 53T^{2} \)
59 \( 1 - 9.11T + 59T^{2} \)
61 \( 1 - 9.32T + 61T^{2} \)
67 \( 1 + 8.16T + 67T^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + 9.40T + 73T^{2} \)
79 \( 1 - 5.94T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 2.69T + 89T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−7.34715967856635467301315790439, −6.79781598264985869351873830053, −5.85385651413934893609022379043, −5.69328145143680377208737852392, −4.68740831093745594489855484068, −4.02314324422394469286851573051, −3.11222572879259596768797212287, −2.26183415164715995717098064441, −1.00703752937106978641895816237, 0, 1.00703752937106978641895816237, 2.26183415164715995717098064441, 3.11222572879259596768797212287, 4.02314324422394469286851573051, 4.68740831093745594489855484068, 5.69328145143680377208737852392, 5.85385651413934893609022379043, 6.79781598264985869351873830053, 7.34715967856635467301315790439

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