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  − 2.82·3-s − 1.92·5-s − 7-s + 4.95·9-s + 11-s + 13-s + 5.41·15-s − 0.965·17-s + 3.09·19-s + 2.82·21-s − 3.72·23-s − 1.30·25-s − 5.51·27-s − 0.786·29-s + 2.78·31-s − 2.82·33-s + 1.92·35-s − 10.8·37-s − 2.82·39-s − 4.59·41-s − 4.94·43-s − 9.52·45-s − 0.799·47-s + 49-s + 2.72·51-s − 1.50·53-s − 1.92·55-s + ⋯
L(s)  = 1  − 1.62·3-s − 0.859·5-s − 0.377·7-s + 1.65·9-s + 0.301·11-s + 0.277·13-s + 1.39·15-s − 0.234·17-s + 0.710·19-s + 0.615·21-s − 0.776·23-s − 0.261·25-s − 1.06·27-s − 0.146·29-s + 0.500·31-s − 0.491·33-s + 0.324·35-s − 1.77·37-s − 0.451·39-s − 0.716·41-s − 0.754·43-s − 1.41·45-s − 0.116·47-s + 0.142·49-s + 0.381·51-s − 0.206·53-s − 0.259·55-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 \( 1 + 2.82T + 3T^{2} \)
5 \( 1 + 1.92T + 5T^{2} \)
17 \( 1 + 0.965T + 17T^{2} \)
19 \( 1 - 3.09T + 19T^{2} \)
23 \( 1 + 3.72T + 23T^{2} \)
29 \( 1 + 0.786T + 29T^{2} \)
31 \( 1 - 2.78T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 + 4.94T + 43T^{2} \)
47 \( 1 + 0.799T + 47T^{2} \)
53 \( 1 + 1.50T + 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 12.8T + 67T^{2} \)
71 \( 1 - 4.44T + 71T^{2} \)
73 \( 1 + 1.16T + 73T^{2} \)
79 \( 1 - 9.23T + 79T^{2} \)
83 \( 1 - 6.28T + 83T^{2} \)
89 \( 1 - 4.46T + 89T^{2} \)
97 \( 1 - 7.50T + 97T^{2} \)
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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

−7.25097818306128926618823476249, −6.70064708272405805283117354958, −6.17449568453184682785326316200, −5.33288313249544019313626167610, −4.88737849779356985617641973236, −3.89258061732563178006546738317, −3.48220380921173053325003659774, −2.01025317676713245820816027077, −0.872745076257913355056231590505, 0, 0.872745076257913355056231590505, 2.01025317676713245820816027077, 3.48220380921173053325003659774, 3.89258061732563178006546738317, 4.88737849779356985617641973236, 5.33288313249544019313626167610, 6.17449568453184682785326316200, 6.70064708272405805283117354958, 7.25097818306128926618823476249

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