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  − 1.53·3-s − 0.216·5-s + 7-s − 0.638·9-s − 11-s − 13-s + 0.333·15-s + 1.18·17-s + 7.88·19-s − 1.53·21-s − 1.22·23-s − 4.95·25-s + 5.59·27-s − 5.20·29-s − 4.80·31-s + 1.53·33-s − 0.216·35-s + 8.61·37-s + 1.53·39-s − 7.95·41-s − 3.91·43-s + 0.138·45-s + 3.51·47-s + 49-s − 1.81·51-s + 6.35·53-s + 0.216·55-s + ⋯
L(s)  = 1  − 0.887·3-s − 0.0969·5-s + 0.377·7-s − 0.212·9-s − 0.301·11-s − 0.277·13-s + 0.0860·15-s + 0.286·17-s + 1.80·19-s − 0.335·21-s − 0.254·23-s − 0.990·25-s + 1.07·27-s − 0.966·29-s − 0.862·31-s + 0.267·33-s − 0.0366·35-s + 1.41·37-s + 0.246·39-s − 1.24·41-s − 0.596·43-s + 0.0206·45-s + 0.512·47-s + 0.142·49-s − 0.254·51-s + 0.873·53-s + 0.0292·55-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 \( 1 + 1.53T + 3T^{2} \)
5 \( 1 + 0.216T + 5T^{2} \)
17 \( 1 - 1.18T + 17T^{2} \)
19 \( 1 - 7.88T + 19T^{2} \)
23 \( 1 + 1.22T + 23T^{2} \)
29 \( 1 + 5.20T + 29T^{2} \)
31 \( 1 + 4.80T + 31T^{2} \)
37 \( 1 - 8.61T + 37T^{2} \)
41 \( 1 + 7.95T + 41T^{2} \)
43 \( 1 + 3.91T + 43T^{2} \)
47 \( 1 - 3.51T + 47T^{2} \)
53 \( 1 - 6.35T + 53T^{2} \)
59 \( 1 + 4.23T + 59T^{2} \)
61 \( 1 + 0.624T + 61T^{2} \)
67 \( 1 - 4.74T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 - 16.1T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 3.48T + 83T^{2} \)
89 \( 1 - 3.11T + 89T^{2} \)
97 \( 1 + 3.26T + 97T^{2} \)
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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

−7.58805058432585366091879361395, −6.77673267604000542403232416396, −5.89884323817435766151762399596, −5.41482580236826043126199489834, −4.95504382427129395971133554382, −3.92777067147023848342098845940, −3.16816816909574413964264847926, −2.16220064808075764746137689642, −1.08946894786672667176524897728, 0, 1.08946894786672667176524897728, 2.16220064808075764746137689642, 3.16816816909574413964264847926, 3.92777067147023848342098845940, 4.95504382427129395971133554382, 5.41482580236826043126199489834, 5.89884323817435766151762399596, 6.77673267604000542403232416396, 7.58805058432585366091879361395

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