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  − 3.33·3-s − 0.131·5-s − 7-s + 8.11·9-s + 11-s − 13-s + 0.437·15-s + 4.35·17-s + 0.765·19-s + 3.33·21-s − 2.35·23-s − 4.98·25-s − 17.0·27-s + 1.55·29-s + 3.44·31-s − 3.33·33-s + 0.131·35-s − 6.91·37-s + 3.33·39-s + 7.35·41-s − 9.25·43-s − 1.06·45-s − 2.35·47-s + 49-s − 14.5·51-s + 6.98·53-s − 0.131·55-s + ⋯
L(s)  = 1  − 1.92·3-s − 0.0587·5-s − 0.377·7-s + 2.70·9-s + 0.301·11-s − 0.277·13-s + 0.113·15-s + 1.05·17-s + 0.175·19-s + 0.727·21-s − 0.490·23-s − 0.996·25-s − 3.28·27-s + 0.287·29-s + 0.617·31-s − 0.580·33-s + 0.0222·35-s − 1.13·37-s + 0.533·39-s + 1.14·41-s − 1.41·43-s − 0.158·45-s − 0.343·47-s + 0.142·49-s − 2.03·51-s + 0.959·53-s − 0.0177·55-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 \( 1 + 3.33T + 3T^{2} \)
5 \( 1 + 0.131T + 5T^{2} \)
17 \( 1 - 4.35T + 17T^{2} \)
19 \( 1 - 0.765T + 19T^{2} \)
23 \( 1 + 2.35T + 23T^{2} \)
29 \( 1 - 1.55T + 29T^{2} \)
31 \( 1 - 3.44T + 31T^{2} \)
37 \( 1 + 6.91T + 37T^{2} \)
41 \( 1 - 7.35T + 41T^{2} \)
43 \( 1 + 9.25T + 43T^{2} \)
47 \( 1 + 2.35T + 47T^{2} \)
53 \( 1 - 6.98T + 53T^{2} \)
59 \( 1 + 0.151T + 59T^{2} \)
61 \( 1 + 7.77T + 61T^{2} \)
67 \( 1 + 7.77T + 67T^{2} \)
71 \( 1 + 0.493T + 71T^{2} \)
73 \( 1 + 3.05T + 73T^{2} \)
79 \( 1 - 1.38T + 79T^{2} \)
83 \( 1 - 4.49T + 83T^{2} \)
89 \( 1 - 6.66T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−7.37197809354705853731954682665, −6.55752207611660099222820121883, −6.10180409252579901639221772528, −5.48285927569792879757029122487, −4.85359003607718243145294888320, −4.10978772463979062468812194410, −3.32053374108001886652616396946, −1.90466366762016866848537460151, −0.982738684798633869226936440534, 0, 0.982738684798633869226936440534, 1.90466366762016866848537460151, 3.32053374108001886652616396946, 4.10978772463979062468812194410, 4.85359003607718243145294888320, 5.48285927569792879757029122487, 6.10180409252579901639221772528, 6.55752207611660099222820121883, 7.37197809354705853731954682665

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