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  − 3.22·3-s + 1.94·5-s + 7-s + 7.38·9-s − 11-s − 13-s − 6.26·15-s + 4.63·17-s − 4.17·19-s − 3.22·21-s + 6.50·23-s − 1.21·25-s − 14.1·27-s + 3.35·29-s − 7.82·31-s + 3.22·33-s + 1.94·35-s + 3.35·37-s + 3.22·39-s − 5.66·41-s − 1.16·43-s + 14.3·45-s + 10.7·47-s + 49-s − 14.9·51-s − 11.5·53-s − 1.94·55-s + ⋯
L(s)  = 1  − 1.86·3-s + 0.869·5-s + 0.377·7-s + 2.46·9-s − 0.301·11-s − 0.277·13-s − 1.61·15-s + 1.12·17-s − 0.958·19-s − 0.703·21-s + 1.35·23-s − 0.243·25-s − 2.71·27-s + 0.622·29-s − 1.40·31-s + 0.560·33-s + 0.328·35-s + 0.551·37-s + 0.515·39-s − 0.885·41-s − 0.178·43-s + 2.14·45-s + 1.56·47-s + 0.142·49-s − 2.09·51-s − 1.59·53-s − 0.262·55-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 \( 1 + 3.22T + 3T^{2} \)
5 \( 1 - 1.94T + 5T^{2} \)
17 \( 1 - 4.63T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 6.50T + 23T^{2} \)
29 \( 1 - 3.35T + 29T^{2} \)
31 \( 1 + 7.82T + 31T^{2} \)
37 \( 1 - 3.35T + 37T^{2} \)
41 \( 1 + 5.66T + 41T^{2} \)
43 \( 1 + 1.16T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 4.96T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + 0.157T + 71T^{2} \)
73 \( 1 - 1.45T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + 4.26T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 6.52T + 97T^{2} \)
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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

−7.27212830246378525609685255145, −6.64758829355381404149655249601, −5.87204420002162782252797797618, −5.58044286939944729772038404236, −4.86209125628582760737052866113, −4.31160115690270318015732451028, −3.08941218201288324075585218942, −1.86420606682495753654733450056, −1.18144728425797136419670867654, 0, 1.18144728425797136419670867654, 1.86420606682495753654733450056, 3.08941218201288324075585218942, 4.31160115690270318015732451028, 4.86209125628582760737052866113, 5.58044286939944729772038404236, 5.87204420002162782252797797618, 6.64758829355381404149655249601, 7.27212830246378525609685255145

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