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  − 1.10·3-s − 2.48·5-s − 7-s − 1.76·9-s + 11-s − 13-s + 2.75·15-s − 6.03·17-s + 0.989·19-s + 1.10·21-s + 8.81·23-s + 1.16·25-s + 5.29·27-s − 4.44·29-s + 2.66·31-s − 1.10·33-s + 2.48·35-s − 10.0·37-s + 1.10·39-s − 1.71·41-s − 12.1·43-s + 4.38·45-s − 2.96·47-s + 49-s + 6.69·51-s − 7.19·53-s − 2.48·55-s + ⋯
L(s)  = 1  − 0.640·3-s − 1.10·5-s − 0.377·7-s − 0.589·9-s + 0.301·11-s − 0.277·13-s + 0.711·15-s − 1.46·17-s + 0.226·19-s + 0.242·21-s + 1.83·23-s + 0.232·25-s + 1.01·27-s − 0.826·29-s + 0.479·31-s − 0.193·33-s + 0.419·35-s − 1.65·37-s + 0.177·39-s − 0.268·41-s − 1.85·43-s + 0.654·45-s − 0.433·47-s + 0.142·49-s + 0.937·51-s − 0.987·53-s − 0.334·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.3082284464$
$L(\frac12)$  $\approx$  $0.3082284464$
$L(\frac{3}{2})$   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 \( 1 + 1.10T + 3T^{2} \)
5 \( 1 + 2.48T + 5T^{2} \)
17 \( 1 + 6.03T + 17T^{2} \)
19 \( 1 - 0.989T + 19T^{2} \)
23 \( 1 - 8.81T + 23T^{2} \)
29 \( 1 + 4.44T + 29T^{2} \)
31 \( 1 - 2.66T + 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 + 1.71T + 41T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + 2.96T + 47T^{2} \)
53 \( 1 + 7.19T + 53T^{2} \)
59 \( 1 + 7.72T + 59T^{2} \)
61 \( 1 + 0.918T + 61T^{2} \)
67 \( 1 - 4.50T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 5.38T + 73T^{2} \)
79 \( 1 + 5.36T + 79T^{2} \)
83 \( 1 + 2.96T + 83T^{2} \)
89 \( 1 - 3.26T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−7.77712247619597559472407326967, −6.87808523335556815780624166363, −6.73071569128102060984230764452, −5.72364437792663385358954340168, −4.94592901789584451386385322008, −4.44439869230457945602410906323, −3.42298508291439579794687011473, −2.93855753711743076370788275495, −1.63673732219487163474573030837, −0.27926487709821426737903592466, 0.27926487709821426737903592466, 1.63673732219487163474573030837, 2.93855753711743076370788275495, 3.42298508291439579794687011473, 4.44439869230457945602410906323, 4.94592901789584451386385322008, 5.72364437792663385358954340168, 6.73071569128102060984230764452, 6.87808523335556815780624166363, 7.77712247619597559472407326967

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