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  + 0.536·3-s − 3.30·5-s − 7-s − 2.71·9-s + 11-s + 13-s − 1.77·15-s + 3.13·17-s − 3.06·19-s − 0.536·21-s + 0.420·23-s + 5.89·25-s − 3.06·27-s + 0.161·29-s + 10.3·31-s + 0.536·33-s + 3.30·35-s + 4.54·37-s + 0.536·39-s − 9.30·41-s + 2.24·43-s + 8.94·45-s + 2.05·47-s + 49-s + 1.68·51-s + 8.66·53-s − 3.30·55-s + ⋯
L(s)  = 1  + 0.309·3-s − 1.47·5-s − 0.377·7-s − 0.903·9-s + 0.301·11-s + 0.277·13-s − 0.457·15-s + 0.759·17-s − 0.702·19-s − 0.117·21-s + 0.0876·23-s + 1.17·25-s − 0.589·27-s + 0.0300·29-s + 1.86·31-s + 0.0934·33-s + 0.557·35-s + 0.747·37-s + 0.0859·39-s − 1.45·41-s + 0.342·43-s + 1.33·45-s + 0.300·47-s + 0.142·49-s + 0.235·51-s + 1.19·53-s − 0.444·55-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 \( 1 - 0.536T + 3T^{2} \)
5 \( 1 + 3.30T + 5T^{2} \)
17 \( 1 - 3.13T + 17T^{2} \)
19 \( 1 + 3.06T + 19T^{2} \)
23 \( 1 - 0.420T + 23T^{2} \)
29 \( 1 - 0.161T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + 9.30T + 41T^{2} \)
43 \( 1 - 2.24T + 43T^{2} \)
47 \( 1 - 2.05T + 47T^{2} \)
53 \( 1 - 8.66T + 53T^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 + 6.87T + 61T^{2} \)
67 \( 1 - 8.01T + 67T^{2} \)
71 \( 1 + 16.2T + 71T^{2} \)
73 \( 1 - 4.62T + 73T^{2} \)
79 \( 1 + 6.36T + 79T^{2} \)
83 \( 1 - 3.20T + 83T^{2} \)
89 \( 1 - 5.49T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.66841611195648886392753834310, −6.83586040696135180162873642541, −6.21819276240802729905519894930, −5.38621689705733445270247381610, −4.45192031708981803783745479257, −3.84442129392224333557781188531, −3.18375683905111387990470460836, −2.50515502191842432925008762267, −1.04709616696701113969506369400, 0, 1.04709616696701113969506369400, 2.50515502191842432925008762267, 3.18375683905111387990470460836, 3.84442129392224333557781188531, 4.45192031708981803783745479257, 5.38621689705733445270247381610, 6.21819276240802729905519894930, 6.83586040696135180162873642541, 7.66841611195648886392753834310

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