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  + 0.979·3-s + 2.33·5-s − 7-s − 2.04·9-s + 11-s − 13-s + 2.28·15-s − 2.60·17-s + 2.89·19-s − 0.979·21-s + 3.81·23-s + 0.442·25-s − 4.93·27-s − 3.10·29-s − 7.32·31-s + 0.979·33-s − 2.33·35-s − 1.46·37-s − 0.979·39-s − 0.286·41-s − 9.21·43-s − 4.76·45-s + 3.81·47-s + 49-s − 2.55·51-s − 11.8·53-s + 2.33·55-s + ⋯
L(s)  = 1  + 0.565·3-s + 1.04·5-s − 0.377·7-s − 0.680·9-s + 0.301·11-s − 0.277·13-s + 0.589·15-s − 0.632·17-s + 0.664·19-s − 0.213·21-s + 0.796·23-s + 0.0884·25-s − 0.950·27-s − 0.577·29-s − 1.31·31-s + 0.170·33-s − 0.394·35-s − 0.240·37-s − 0.156·39-s − 0.0446·41-s − 1.40·43-s − 0.709·45-s + 0.556·47-s + 0.142·49-s − 0.357·51-s − 1.63·53-s + 0.314·55-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 \( 1 - 0.979T + 3T^{2} \)
5 \( 1 - 2.33T + 5T^{2} \)
17 \( 1 + 2.60T + 17T^{2} \)
19 \( 1 - 2.89T + 19T^{2} \)
23 \( 1 - 3.81T + 23T^{2} \)
29 \( 1 + 3.10T + 29T^{2} \)
31 \( 1 + 7.32T + 31T^{2} \)
37 \( 1 + 1.46T + 37T^{2} \)
41 \( 1 + 0.286T + 41T^{2} \)
43 \( 1 + 9.21T + 43T^{2} \)
47 \( 1 - 3.81T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 + 9.80T + 59T^{2} \)
61 \( 1 - 1.26T + 61T^{2} \)
67 \( 1 + 4.93T + 67T^{2} \)
71 \( 1 - 16.2T + 71T^{2} \)
73 \( 1 - 0.183T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 15.3T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 4.70T + 97T^{2} \)
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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

−7.45753058592537110654509617045, −6.79525974478163531714386117205, −6.06114718515214039058045325756, −5.48103964624527344603833234330, −4.79202904175615038311395818320, −3.65807005931834902160080358334, −3.07280960868140068141538114119, −2.24097641318711136695959758480, −1.52745346946479709337911809789, 0, 1.52745346946479709337911809789, 2.24097641318711136695959758480, 3.07280960868140068141538114119, 3.65807005931834902160080358334, 4.79202904175615038311395818320, 5.48103964624527344603833234330, 6.06114718515214039058045325756, 6.79525974478163531714386117205, 7.45753058592537110654509617045

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