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  − 0.528·3-s + 0.933·5-s + 7-s − 2.72·9-s − 11-s − 13-s − 0.493·15-s + 0.372·17-s − 3.48·19-s − 0.528·21-s − 4.94·23-s − 4.12·25-s + 3.02·27-s + 9.77·29-s + 8.42·31-s + 0.528·33-s + 0.933·35-s − 2.32·37-s + 0.528·39-s + 8.25·41-s + 1.29·43-s − 2.54·45-s + 10.0·47-s + 49-s − 0.196·51-s + 1.68·53-s − 0.933·55-s + ⋯
L(s)  = 1  − 0.305·3-s + 0.417·5-s + 0.377·7-s − 0.906·9-s − 0.301·11-s − 0.277·13-s − 0.127·15-s + 0.0902·17-s − 0.798·19-s − 0.115·21-s − 1.03·23-s − 0.825·25-s + 0.582·27-s + 1.81·29-s + 1.51·31-s + 0.0920·33-s + 0.157·35-s − 0.381·37-s + 0.0846·39-s + 1.28·41-s + 0.197·43-s − 0.378·45-s + 1.46·47-s + 0.142·49-s − 0.0275·51-s + 0.230·53-s − 0.125·55-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 \( 1 + 0.528T + 3T^{2} \)
5 \( 1 - 0.933T + 5T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 + 3.48T + 19T^{2} \)
23 \( 1 + 4.94T + 23T^{2} \)
29 \( 1 - 9.77T + 29T^{2} \)
31 \( 1 - 8.42T + 31T^{2} \)
37 \( 1 + 2.32T + 37T^{2} \)
41 \( 1 - 8.25T + 41T^{2} \)
43 \( 1 - 1.29T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 1.68T + 53T^{2} \)
59 \( 1 - 13.7T + 59T^{2} \)
61 \( 1 + 6.52T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + 9.57T + 71T^{2} \)
73 \( 1 + 3.18T + 73T^{2} \)
79 \( 1 + 17.5T + 79T^{2} \)
83 \( 1 + 2.45T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + 6.79T + 97T^{2} \)
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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

−7.53275365164324855665687720537, −6.68660214580312491874438502051, −5.89692603207504055247203582393, −5.66657410174784872018501670798, −4.60045910738049366441115992440, −4.14357099756275777018877441698, −2.79718030699381078582106173845, −2.42343407979205093103505381644, −1.21392015727861228154990853987, 0, 1.21392015727861228154990853987, 2.42343407979205093103505381644, 2.79718030699381078582106173845, 4.14357099756275777018877441698, 4.60045910738049366441115992440, 5.66657410174784872018501670798, 5.89692603207504055247203582393, 6.68660214580312491874438502051, 7.53275365164324855665687720537

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