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  − 0.830·3-s + 3.93·5-s + 7-s − 2.31·9-s − 11-s − 13-s − 3.26·15-s + 7.93·17-s + 4.36·19-s − 0.830·21-s + 1.83·23-s + 10.4·25-s + 4.41·27-s − 4.44·29-s + 1.61·31-s + 0.830·33-s + 3.93·35-s + 5.72·37-s + 0.830·39-s + 4.67·41-s − 1.06·43-s − 9.08·45-s + 2.25·47-s + 49-s − 6.58·51-s − 3.24·53-s − 3.93·55-s + ⋯
L(s)  = 1  − 0.479·3-s + 1.75·5-s + 0.377·7-s − 0.770·9-s − 0.301·11-s − 0.277·13-s − 0.842·15-s + 1.92·17-s + 1.00·19-s − 0.181·21-s + 0.382·23-s + 2.09·25-s + 0.848·27-s − 0.824·29-s + 0.290·31-s + 0.144·33-s + 0.664·35-s + 0.940·37-s + 0.132·39-s + 0.729·41-s − 0.162·43-s − 1.35·45-s + 0.328·47-s + 0.142·49-s − 0.922·51-s − 0.445·53-s − 0.530·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.734033514$
$L(\frac12)$  $\approx$  $2.734033514$
$L(\frac{3}{2})$   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 \( 1 + 0.830T + 3T^{2} \)
5 \( 1 - 3.93T + 5T^{2} \)
17 \( 1 - 7.93T + 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 - 1.83T + 23T^{2} \)
29 \( 1 + 4.44T + 29T^{2} \)
31 \( 1 - 1.61T + 31T^{2} \)
37 \( 1 - 5.72T + 37T^{2} \)
41 \( 1 - 4.67T + 41T^{2} \)
43 \( 1 + 1.06T + 43T^{2} \)
47 \( 1 - 2.25T + 47T^{2} \)
53 \( 1 + 3.24T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 - 1.90T + 61T^{2} \)
67 \( 1 + 3.90T + 67T^{2} \)
71 \( 1 - 8.09T + 71T^{2} \)
73 \( 1 + 8.62T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 0.0260T + 89T^{2} \)
97 \( 1 + 1.13T + 97T^{2} \)
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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

−7.75449012926567563807634549789, −7.14105783135347348417597638150, −6.10273509727894688984599640423, −5.70893024481085823932200373214, −5.35047726372853123138548497637, −4.60538966098885951867697288679, −3.19836531745547061472953756429, −2.71397492234039464088391002515, −1.67088108786584385522571126638, −0.890433851686605851861420779703, 0.890433851686605851861420779703, 1.67088108786584385522571126638, 2.71397492234039464088391002515, 3.19836531745547061472953756429, 4.60538966098885951867697288679, 5.35047726372853123138548497637, 5.70893024481085823932200373214, 6.10273509727894688984599640423, 7.14105783135347348417597638150, 7.75449012926567563807634549789

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