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  − 0.447·3-s + 3.28·5-s − 7-s − 2.79·9-s + 11-s − 13-s − 1.46·15-s + 4.02·17-s + 1.40·19-s + 0.447·21-s − 4.52·23-s + 5.76·25-s + 2.59·27-s − 1.61·29-s − 2.93·31-s − 0.447·33-s − 3.28·35-s − 11.8·37-s + 0.447·39-s − 6.65·41-s − 3.31·43-s − 9.18·45-s − 4.52·47-s + 49-s − 1.80·51-s + 10.5·53-s + 3.28·55-s + ⋯
L(s)  = 1  − 0.258·3-s + 1.46·5-s − 0.377·7-s − 0.933·9-s + 0.301·11-s − 0.277·13-s − 0.379·15-s + 0.976·17-s + 0.322·19-s + 0.0977·21-s − 0.943·23-s + 1.15·25-s + 0.499·27-s − 0.300·29-s − 0.526·31-s − 0.0779·33-s − 0.554·35-s − 1.94·37-s + 0.0717·39-s − 1.03·41-s − 0.505·43-s − 1.36·45-s − 0.659·47-s + 0.142·49-s − 0.252·51-s + 1.45·53-s + 0.442·55-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 \( 1 + 0.447T + 3T^{2} \)
5 \( 1 - 3.28T + 5T^{2} \)
17 \( 1 - 4.02T + 17T^{2} \)
19 \( 1 - 1.40T + 19T^{2} \)
23 \( 1 + 4.52T + 23T^{2} \)
29 \( 1 + 1.61T + 29T^{2} \)
31 \( 1 + 2.93T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 6.65T + 41T^{2} \)
43 \( 1 + 3.31T + 43T^{2} \)
47 \( 1 + 4.52T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 5.53T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 4.73T + 67T^{2} \)
71 \( 1 + 9.78T + 71T^{2} \)
73 \( 1 + 4.97T + 73T^{2} \)
79 \( 1 + 0.859T + 79T^{2} \)
83 \( 1 + 17.7T + 83T^{2} \)
89 \( 1 - 3.79T + 89T^{2} \)
97 \( 1 + 18.7T + 97T^{2} \)
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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

−7.31421633811659085715073688041, −6.69104871066129675891386502716, −5.93532778166946293047724619068, −5.53726202942222912287308154578, −5.01382518336802593534388931102, −3.74465356099404754866457515551, −3.05737404803121980803728707653, −2.16213816027521598313372587846, −1.40119889344467291967683755532, 0, 1.40119889344467291967683755532, 2.16213816027521598313372587846, 3.05737404803121980803728707653, 3.74465356099404754866457515551, 5.01382518336802593534388931102, 5.53726202942222912287308154578, 5.93532778166946293047724619068, 6.69104871066129675891386502716, 7.31421633811659085715073688041

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