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  − 1.64·3-s + 2.39·5-s − 7-s − 0.301·9-s − 11-s − 13-s − 3.94·15-s − 5.56·17-s + 1.34·19-s + 1.64·21-s + 2.38·23-s + 0.758·25-s + 5.42·27-s − 0.520·29-s + 8.43·31-s + 1.64·33-s − 2.39·35-s − 5.34·37-s + 1.64·39-s + 1.51·41-s + 8.44·43-s − 0.722·45-s − 0.408·47-s + 49-s + 9.13·51-s + 14.0·53-s − 2.39·55-s + ⋯
L(s)  = 1  − 0.948·3-s + 1.07·5-s − 0.377·7-s − 0.100·9-s − 0.301·11-s − 0.277·13-s − 1.01·15-s − 1.34·17-s + 0.308·19-s + 0.358·21-s + 0.497·23-s + 0.151·25-s + 1.04·27-s − 0.0965·29-s + 1.51·31-s + 0.285·33-s − 0.405·35-s − 0.879·37-s + 0.263·39-s + 0.236·41-s + 1.28·43-s − 0.107·45-s − 0.0595·47-s + 0.142·49-s + 1.27·51-s + 1.93·53-s − 0.323·55-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 \( 1 + 1.64T + 3T^{2} \)
5 \( 1 - 2.39T + 5T^{2} \)
17 \( 1 + 5.56T + 17T^{2} \)
19 \( 1 - 1.34T + 19T^{2} \)
23 \( 1 - 2.38T + 23T^{2} \)
29 \( 1 + 0.520T + 29T^{2} \)
31 \( 1 - 8.43T + 31T^{2} \)
37 \( 1 + 5.34T + 37T^{2} \)
41 \( 1 - 1.51T + 41T^{2} \)
43 \( 1 - 8.44T + 43T^{2} \)
47 \( 1 + 0.408T + 47T^{2} \)
53 \( 1 - 14.0T + 53T^{2} \)
59 \( 1 + 14.2T + 59T^{2} \)
61 \( 1 - 5.50T + 61T^{2} \)
67 \( 1 + 5.69T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 + 16.5T + 79T^{2} \)
83 \( 1 + 5.85T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + 6.66T + 97T^{2} \)
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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

−7.18479309924300091703951980756, −6.66819530048293680262416656066, −6.03162207653685069315989974348, −5.51970800897385692183558890490, −4.88214470244815746123843259396, −4.09874224102434783078593517560, −2.83621684703879289783438041372, −2.32571504312232413714910215847, −1.12970028524986526897660205189, 0, 1.12970028524986526897660205189, 2.32571504312232413714910215847, 2.83621684703879289783438041372, 4.09874224102434783078593517560, 4.88214470244815746123843259396, 5.51970800897385692183558890490, 6.03162207653685069315989974348, 6.66819530048293680262416656066, 7.18479309924300091703951980756

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