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  + 0.129·3-s − 0.197·5-s + 7-s − 2.98·9-s − 11-s − 13-s − 0.0255·15-s + 0.463·17-s + 3.51·19-s + 0.129·21-s + 8.53·23-s − 4.96·25-s − 0.773·27-s − 7.30·29-s + 9.67·31-s − 0.129·33-s − 0.197·35-s − 11.6·37-s − 0.129·39-s + 3.57·41-s − 0.592·43-s + 0.590·45-s − 3.21·47-s + 49-s + 0.0599·51-s − 11.5·53-s + 0.197·55-s + ⋯
L(s)  = 1  + 0.0746·3-s − 0.0884·5-s + 0.377·7-s − 0.994·9-s − 0.301·11-s − 0.277·13-s − 0.00660·15-s + 0.112·17-s + 0.807·19-s + 0.0282·21-s + 1.77·23-s − 0.992·25-s − 0.148·27-s − 1.35·29-s + 1.73·31-s − 0.0224·33-s − 0.0334·35-s − 1.91·37-s − 0.0206·39-s + 0.557·41-s − 0.0903·43-s + 0.0879·45-s − 0.468·47-s + 0.142·49-s + 0.00839·51-s − 1.58·53-s + 0.0266·55-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 \( 1 - 0.129T + 3T^{2} \)
5 \( 1 + 0.197T + 5T^{2} \)
17 \( 1 - 0.463T + 17T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
23 \( 1 - 8.53T + 23T^{2} \)
29 \( 1 + 7.30T + 29T^{2} \)
31 \( 1 - 9.67T + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 + 0.592T + 43T^{2} \)
47 \( 1 + 3.21T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 4.83T + 61T^{2} \)
67 \( 1 - 6.38T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 - 3.76T + 73T^{2} \)
79 \( 1 + 3.12T + 79T^{2} \)
83 \( 1 + 4.06T + 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 6.85T + 97T^{2} \)
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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

−7.67273235860972567145451607249, −6.81551345047661543938284180750, −6.09363584647733630644498181984, −5.12659106761004100532934665753, −5.03508329286379739180488976022, −3.74902115018937030104497541446, −3.10578479460651291881400429938, −2.32685614308676805244500595038, −1.25031345803437996778757575666, 0, 1.25031345803437996778757575666, 2.32685614308676805244500595038, 3.10578479460651291881400429938, 3.74902115018937030104497541446, 5.03508329286379739180488976022, 5.12659106761004100532934665753, 6.09363584647733630644498181984, 6.81551345047661543938284180750, 7.67273235860972567145451607249

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