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  − 1.00·3-s + 2.49·5-s − 7-s − 1.98·9-s + 11-s + 13-s − 2.52·15-s − 4.53·17-s − 0.365·19-s + 1.00·21-s + 2.18·23-s + 1.23·25-s + 5.02·27-s + 7.70·29-s − 6.21·31-s − 1.00·33-s − 2.49·35-s − 6.41·37-s − 1.00·39-s − 0.881·41-s − 2.18·43-s − 4.94·45-s + 13.5·47-s + 49-s + 4.57·51-s − 5.98·53-s + 2.49·55-s + ⋯
L(s)  = 1  − 0.582·3-s + 1.11·5-s − 0.377·7-s − 0.660·9-s + 0.301·11-s + 0.277·13-s − 0.651·15-s − 1.09·17-s − 0.0839·19-s + 0.220·21-s + 0.455·23-s + 0.247·25-s + 0.967·27-s + 1.43·29-s − 1.11·31-s − 0.175·33-s − 0.422·35-s − 1.05·37-s − 0.161·39-s − 0.137·41-s − 0.332·43-s − 0.737·45-s + 1.97·47-s + 0.142·49-s + 0.641·51-s − 0.821·53-s + 0.336·55-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 \( 1 + 1.00T + 3T^{2} \)
5 \( 1 - 2.49T + 5T^{2} \)
17 \( 1 + 4.53T + 17T^{2} \)
19 \( 1 + 0.365T + 19T^{2} \)
23 \( 1 - 2.18T + 23T^{2} \)
29 \( 1 - 7.70T + 29T^{2} \)
31 \( 1 + 6.21T + 31T^{2} \)
37 \( 1 + 6.41T + 37T^{2} \)
41 \( 1 + 0.881T + 41T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 - 13.5T + 47T^{2} \)
53 \( 1 + 5.98T + 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 4.84T + 61T^{2} \)
67 \( 1 + 0.801T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 - 2.15T + 89T^{2} \)
97 \( 1 + 4.92T + 97T^{2} \)
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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

−7.18823707481763752534769756140, −6.67543422337086333898482664794, −6.04467148776705843815215991050, −5.56061393060079526550361963313, −4.85445266321019468983192437644, −3.96102693417568908188611355585, −2.95035979390530377935584761874, −2.24681862179079084309703973190, −1.24853594261325381188601390313, 0, 1.24853594261325381188601390313, 2.24681862179079084309703973190, 2.95035979390530377935584761874, 3.96102693417568908188611355585, 4.85445266321019468983192437644, 5.56061393060079526550361963313, 6.04467148776705843815215991050, 6.67543422337086333898482664794, 7.18823707481763752534769756140

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