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  − 0.287·3-s − 0.115·5-s + 7-s − 2.91·9-s − 11-s − 13-s + 0.0331·15-s − 5.96·17-s + 2.56·19-s − 0.287·21-s + 1.63·23-s − 4.98·25-s + 1.69·27-s − 4.27·29-s + 9.90·31-s + 0.287·33-s − 0.115·35-s + 3.82·37-s + 0.287·39-s − 6.84·41-s + 2.08·43-s + 0.336·45-s − 10.6·47-s + 49-s + 1.71·51-s + 13.3·53-s + 0.115·55-s + ⋯
L(s)  = 1  − 0.165·3-s − 0.0515·5-s + 0.377·7-s − 0.972·9-s − 0.301·11-s − 0.277·13-s + 0.00854·15-s − 1.44·17-s + 0.587·19-s − 0.0626·21-s + 0.340·23-s − 0.997·25-s + 0.327·27-s − 0.793·29-s + 1.77·31-s + 0.0500·33-s − 0.0194·35-s + 0.629·37-s + 0.0459·39-s − 1.06·41-s + 0.317·43-s + 0.0501·45-s − 1.55·47-s + 0.142·49-s + 0.239·51-s + 1.83·53-s + 0.0155·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.225740629$
$L(\frac12)$  $\approx$  $1.225740629$
$L(\frac{3}{2})$   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 \( 1 + 0.287T + 3T^{2} \)
5 \( 1 + 0.115T + 5T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 - 2.56T + 19T^{2} \)
23 \( 1 - 1.63T + 23T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 - 9.90T + 31T^{2} \)
37 \( 1 - 3.82T + 37T^{2} \)
41 \( 1 + 6.84T + 41T^{2} \)
43 \( 1 - 2.08T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 - 12.8T + 59T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 6.75T + 67T^{2} \)
71 \( 1 + 6.49T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + 3.16T + 79T^{2} \)
83 \( 1 - 2.81T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−7.79691982197706446658471107866, −7.23372633077182900042622144952, −6.32083975183587140331654591106, −5.83459131102295828766725505075, −4.95131408776866685346831099511, −4.48466770163022266044523523515, −3.43497087503946330687466578912, −2.64070180904266859657864631726, −1.87719523238672362857946798080, −0.53188743677340069151099207308, 0.53188743677340069151099207308, 1.87719523238672362857946798080, 2.64070180904266859657864631726, 3.43497087503946330687466578912, 4.48466770163022266044523523515, 4.95131408776866685346831099511, 5.83459131102295828766725505075, 6.32083975183587140331654591106, 7.23372633077182900042622144952, 7.79691982197706446658471107866

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