Properties

Degree 2
Conductor $ 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.00·3-s + 2.62·5-s − 7-s + 6.03·9-s + 11-s − 13-s − 7.88·15-s − 0.640·17-s − 5.26·19-s + 3.00·21-s + 3.73·23-s + 1.87·25-s − 9.13·27-s − 2.10·29-s − 10.8·31-s − 3.00·33-s − 2.62·35-s − 5.89·37-s + 3.00·39-s + 10.7·41-s + 2.17·43-s + 15.8·45-s + 3.39·47-s + 49-s + 1.92·51-s + 3.38·53-s + 2.62·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.17·5-s − 0.377·7-s + 2.01·9-s + 0.301·11-s − 0.277·13-s − 2.03·15-s − 0.155·17-s − 1.20·19-s + 0.656·21-s + 0.779·23-s + 0.374·25-s − 1.75·27-s − 0.391·29-s − 1.94·31-s − 0.523·33-s − 0.443·35-s − 0.969·37-s + 0.481·39-s + 1.67·41-s + 0.332·43-s + 2.36·45-s + 0.495·47-s + 0.142·49-s + 0.269·51-s + 0.465·53-s + 0.353·55-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  =  0
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.005678609$
$L(\frac12)$  $\approx$  $1.005678609$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 + 3.00T + 3T^{2} \)
5 \( 1 - 2.62T + 5T^{2} \)
17 \( 1 + 0.640T + 17T^{2} \)
19 \( 1 + 5.26T + 19T^{2} \)
23 \( 1 - 3.73T + 23T^{2} \)
29 \( 1 + 2.10T + 29T^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + 5.89T + 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 2.17T + 43T^{2} \)
47 \( 1 - 3.39T + 47T^{2} \)
53 \( 1 - 3.38T + 53T^{2} \)
59 \( 1 + 3.21T + 59T^{2} \)
61 \( 1 - 14.8T + 61T^{2} \)
67 \( 1 - 7.51T + 67T^{2} \)
71 \( 1 + 2.60T + 71T^{2} \)
73 \( 1 + 0.811T + 73T^{2} \)
79 \( 1 - 9.43T + 79T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + 8.41T + 89T^{2} \)
97 \( 1 - 8.56T + 97T^{2} \)
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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

−7.44371738251273376831023283453, −6.89088426969203826111003670674, −6.35247355800254782380362709808, −5.63002990947456119033844153012, −5.42662721491819525374714149262, −4.48122190958619329935750489971, −3.75680556402508006502116970654, −2.38959730801423049334865395645, −1.64102078710959124031392224990, −0.54862557925401252143866036642, 0.54862557925401252143866036642, 1.64102078710959124031392224990, 2.38959730801423049334865395645, 3.75680556402508006502116970654, 4.48122190958619329935750489971, 5.42662721491819525374714149262, 5.63002990947456119033844153012, 6.35247355800254782380362709808, 6.89088426969203826111003670674, 7.44371738251273376831023283453

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