Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.49·3-s − 2.57·5-s − 7-s − 0.767·9-s + 11-s − 13-s + 3.85·15-s − 3.29·17-s + 3.91·19-s + 1.49·21-s − 2.83·23-s + 1.65·25-s + 5.62·27-s − 5.39·29-s + 7.50·31-s − 1.49·33-s + 2.57·35-s + 8.01·37-s + 1.49·39-s − 4.23·41-s − 2.91·43-s + 1.97·45-s − 2.83·47-s + 49-s + 4.92·51-s + 2.31·53-s − 2.57·55-s + ⋯
L(s)  = 1  − 0.862·3-s − 1.15·5-s − 0.377·7-s − 0.255·9-s + 0.301·11-s − 0.277·13-s + 0.995·15-s − 0.799·17-s + 0.898·19-s + 0.326·21-s − 0.591·23-s + 0.330·25-s + 1.08·27-s − 1.00·29-s + 1.34·31-s − 0.260·33-s + 0.436·35-s + 1.31·37-s + 0.239·39-s − 0.661·41-s − 0.444·43-s + 0.295·45-s − 0.413·47-s + 0.142·49-s + 0.689·51-s + 0.317·53-s − 0.347·55-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  =  1
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 1.49T + 3T^{2} \)
5 \( 1 + 2.57T + 5T^{2} \)
17 \( 1 + 3.29T + 17T^{2} \)
19 \( 1 - 3.91T + 19T^{2} \)
23 \( 1 + 2.83T + 23T^{2} \)
29 \( 1 + 5.39T + 29T^{2} \)
31 \( 1 - 7.50T + 31T^{2} \)
37 \( 1 - 8.01T + 37T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 + 2.83T + 47T^{2} \)
53 \( 1 - 2.31T + 53T^{2} \)
59 \( 1 + 5.37T + 59T^{2} \)
61 \( 1 - 0.473T + 61T^{2} \)
67 \( 1 - 4.81T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 8.61T + 73T^{2} \)
79 \( 1 + 1.69T + 79T^{2} \)
83 \( 1 - 5.19T + 83T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 - 5.14T + 97T^{2} \)
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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

−7.56385052297946924073589537782, −6.60268102210326734901821991491, −6.28590186211695820601275525258, −5.33157919581151684362711124433, −4.72351660314330747472766238496, −3.93758824925527423211654477011, −3.26507242411345643221344155955, −2.28451496244898141683457338095, −0.865145367915235029067758848789, 0, 0.865145367915235029067758848789, 2.28451496244898141683457338095, 3.26507242411345643221344155955, 3.93758824925527423211654477011, 4.72351660314330747472766238496, 5.33157919581151684362711124433, 6.28590186211695820601275525258, 6.60268102210326734901821991491, 7.56385052297946924073589537782

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