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  + 3.22·3-s − 3.58·5-s + 7-s + 7.37·9-s − 11-s − 13-s − 11.5·15-s − 4.82·17-s − 7.88·19-s + 3.22·21-s + 4.80·23-s + 7.87·25-s + 14.0·27-s + 9.62·29-s − 5.44·31-s − 3.22·33-s − 3.58·35-s + 0.187·37-s − 3.22·39-s + 1.50·41-s − 9.53·43-s − 26.4·45-s − 3.00·47-s + 49-s − 15.5·51-s − 2.34·53-s + 3.58·55-s + ⋯
L(s)  = 1  + 1.85·3-s − 1.60·5-s + 0.377·7-s + 2.45·9-s − 0.301·11-s − 0.277·13-s − 2.98·15-s − 1.16·17-s − 1.80·19-s + 0.702·21-s + 1.00·23-s + 1.57·25-s + 2.70·27-s + 1.78·29-s − 0.978·31-s − 0.560·33-s − 0.606·35-s + 0.0309·37-s − 0.515·39-s + 0.235·41-s − 1.45·43-s − 3.94·45-s − 0.438·47-s + 0.142·49-s − 2.17·51-s − 0.322·53-s + 0.483·55-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 \( 1 - 3.22T + 3T^{2} \)
5 \( 1 + 3.58T + 5T^{2} \)
17 \( 1 + 4.82T + 17T^{2} \)
19 \( 1 + 7.88T + 19T^{2} \)
23 \( 1 - 4.80T + 23T^{2} \)
29 \( 1 - 9.62T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 - 0.187T + 37T^{2} \)
41 \( 1 - 1.50T + 41T^{2} \)
43 \( 1 + 9.53T + 43T^{2} \)
47 \( 1 + 3.00T + 47T^{2} \)
53 \( 1 + 2.34T + 53T^{2} \)
59 \( 1 + 4.59T + 59T^{2} \)
61 \( 1 - 5.05T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 9.44T + 89T^{2} \)
97 \( 1 + 0.672T + 97T^{2} \)
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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

−7.72951509789919819134626771411, −6.99674982404976231110721390413, −6.59054727292721150241187020206, −4.89173211323678543935472747554, −4.41028910123918844220623856012, −3.90277540842764211455230270263, −3.04347141026417679497029286901, −2.49383204906576317536015207055, −1.51140627665835562867639780584, 0, 1.51140627665835562867639780584, 2.49383204906576317536015207055, 3.04347141026417679497029286901, 3.90277540842764211455230270263, 4.41028910123918844220623856012, 4.89173211323678543935472747554, 6.59054727292721150241187020206, 6.99674982404976231110721390413, 7.72951509789919819134626771411

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