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  + 0.398·3-s + 3.53·5-s − 7-s − 2.84·9-s − 11-s − 13-s + 1.40·15-s − 1.07·17-s + 5.12·19-s − 0.398·21-s − 5.71·23-s + 7.46·25-s − 2.32·27-s − 1.17·29-s + 1.41·31-s − 0.398·33-s − 3.53·35-s − 7.80·37-s − 0.398·39-s − 5.27·41-s − 2.86·43-s − 10.0·45-s − 2.62·47-s + 49-s − 0.429·51-s − 6.09·53-s − 3.53·55-s + ⋯
L(s)  = 1  + 0.229·3-s + 1.57·5-s − 0.377·7-s − 0.947·9-s − 0.301·11-s − 0.277·13-s + 0.362·15-s − 0.261·17-s + 1.17·19-s − 0.0868·21-s − 1.19·23-s + 1.49·25-s − 0.447·27-s − 0.218·29-s + 0.253·31-s − 0.0693·33-s − 0.596·35-s − 1.28·37-s − 0.0637·39-s − 0.824·41-s − 0.437·43-s − 1.49·45-s − 0.383·47-s + 0.142·49-s − 0.0601·51-s − 0.837·53-s − 0.476·55-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 \( 1 - 0.398T + 3T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
17 \( 1 + 1.07T + 17T^{2} \)
19 \( 1 - 5.12T + 19T^{2} \)
23 \( 1 + 5.71T + 23T^{2} \)
29 \( 1 + 1.17T + 29T^{2} \)
31 \( 1 - 1.41T + 31T^{2} \)
37 \( 1 + 7.80T + 37T^{2} \)
41 \( 1 + 5.27T + 41T^{2} \)
43 \( 1 + 2.86T + 43T^{2} \)
47 \( 1 + 2.62T + 47T^{2} \)
53 \( 1 + 6.09T + 53T^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 + 1.98T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 8.91T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 - 2.00T + 83T^{2} \)
89 \( 1 + 5.89T + 89T^{2} \)
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

−7.47999279962735786814148748844, −6.62236974221753751616498953535, −6.08645914363649282098564586830, −5.40445970452034560379379648469, −5.00504435940209845845290836152, −3.70613522673688058138437502169, −2.94452849904266848381972655043, −2.27400241457980682878498697905, −1.49167266948210717582785077407, 0, 1.49167266948210717582785077407, 2.27400241457980682878498697905, 2.94452849904266848381972655043, 3.70613522673688058138437502169, 5.00504435940209845845290836152, 5.40445970452034560379379648469, 6.08645914363649282098564586830, 6.62236974221753751616498953535, 7.47999279962735786814148748844

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