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  + 3.25·3-s − 0.215·5-s + 7-s + 7.56·9-s − 11-s − 13-s − 0.701·15-s − 1.35·17-s + 5.10·19-s + 3.25·21-s + 4.33·23-s − 4.95·25-s + 14.8·27-s − 0.702·29-s + 9.03·31-s − 3.25·33-s − 0.215·35-s − 2.18·37-s − 3.25·39-s + 3.79·41-s + 2.91·43-s − 1.63·45-s + 10.5·47-s + 49-s − 4.40·51-s − 8.94·53-s + 0.215·55-s + ⋯
L(s)  = 1  + 1.87·3-s − 0.0964·5-s + 0.377·7-s + 2.52·9-s − 0.301·11-s − 0.277·13-s − 0.181·15-s − 0.328·17-s + 1.17·19-s + 0.709·21-s + 0.903·23-s − 0.990·25-s + 2.85·27-s − 0.130·29-s + 1.62·31-s − 0.565·33-s − 0.0364·35-s − 0.359·37-s − 0.520·39-s + 0.593·41-s + 0.444·43-s − 0.243·45-s + 1.54·47-s + 0.142·49-s − 0.616·51-s − 1.22·53-s + 0.0290·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.836108541$
$L(\frac12)$  $\approx$  $4.836108541$
$L(\frac{3}{2})$   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 \( 1 - 3.25T + 3T^{2} \)
5 \( 1 + 0.215T + 5T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 - 5.10T + 19T^{2} \)
23 \( 1 - 4.33T + 23T^{2} \)
29 \( 1 + 0.702T + 29T^{2} \)
31 \( 1 - 9.03T + 31T^{2} \)
37 \( 1 + 2.18T + 37T^{2} \)
41 \( 1 - 3.79T + 41T^{2} \)
43 \( 1 - 2.91T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 8.94T + 53T^{2} \)
59 \( 1 + 6.12T + 59T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 - 6.89T + 89T^{2} \)
97 \( 1 + 9.18T + 97T^{2} \)
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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

−7.80401125873373751925659653267, −7.49342213964815289267456376996, −6.76734113468614518516780650562, −5.71617498899189024550160309427, −4.71760898486404074812670148238, −4.22385966540770329732155044031, −3.27270323185389540936114166455, −2.77861199621897909463387438681, −1.99091529368464379479966126902, −1.05192609459721046642260147867, 1.05192609459721046642260147867, 1.99091529368464379479966126902, 2.77861199621897909463387438681, 3.27270323185389540936114166455, 4.22385966540770329732155044031, 4.71760898486404074812670148238, 5.71617498899189024550160309427, 6.76734113468614518516780650562, 7.49342213964815289267456376996, 7.80401125873373751925659653267

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