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  + 1.54·3-s − 0.356·5-s − 7-s − 0.600·9-s − 11-s − 13-s − 0.552·15-s − 6.04·17-s + 8.48·19-s − 1.54·21-s + 7.98·23-s − 4.87·25-s − 5.57·27-s + 5.31·29-s − 9.77·31-s − 1.54·33-s + 0.356·35-s + 10.5·37-s − 1.54·39-s + 8.72·41-s − 7.46·43-s + 0.214·45-s + 1.45·47-s + 49-s − 9.37·51-s − 1.71·53-s + 0.356·55-s + ⋯
L(s)  = 1  + 0.894·3-s − 0.159·5-s − 0.377·7-s − 0.200·9-s − 0.301·11-s − 0.277·13-s − 0.142·15-s − 1.46·17-s + 1.94·19-s − 0.338·21-s + 1.66·23-s − 0.974·25-s − 1.07·27-s + 0.987·29-s − 1.75·31-s − 0.269·33-s + 0.0603·35-s + 1.73·37-s − 0.248·39-s + 1.36·41-s − 1.13·43-s + 0.0319·45-s + 0.212·47-s + 0.142·49-s − 1.31·51-s − 0.235·53-s + 0.0481·55-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 \( 1 - 1.54T + 3T^{2} \)
5 \( 1 + 0.356T + 5T^{2} \)
17 \( 1 + 6.04T + 17T^{2} \)
19 \( 1 - 8.48T + 19T^{2} \)
23 \( 1 - 7.98T + 23T^{2} \)
29 \( 1 - 5.31T + 29T^{2} \)
31 \( 1 + 9.77T + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 - 8.72T + 41T^{2} \)
43 \( 1 + 7.46T + 43T^{2} \)
47 \( 1 - 1.45T + 47T^{2} \)
53 \( 1 + 1.71T + 53T^{2} \)
59 \( 1 + 0.781T + 59T^{2} \)
61 \( 1 + 5.31T + 61T^{2} \)
67 \( 1 + 6.19T + 67T^{2} \)
71 \( 1 + 7.65T + 71T^{2} \)
73 \( 1 + 2.44T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + 5.43T + 83T^{2} \)
89 \( 1 - 9.25T + 89T^{2} \)
97 \( 1 - 3.82T + 97T^{2} \)
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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

−7.51256788851546266981464638341, −7.05442874029935066663612203746, −6.08926878088758594031440321289, −5.39329961839413794906042713564, −4.59374936830261858522298320030, −3.75789773961424168077913420177, −2.91481707441893768589152126901, −2.57731296082003771467397346684, −1.35214261081046446943092972678, 0, 1.35214261081046446943092972678, 2.57731296082003771467397346684, 2.91481707441893768589152126901, 3.75789773961424168077913420177, 4.59374936830261858522298320030, 5.39329961839413794906042713564, 6.08926878088758594031440321289, 7.05442874029935066663612203746, 7.51256788851546266981464638341

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