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  + 2.67·3-s + 0.943·5-s − 7-s + 4.14·9-s − 11-s − 13-s + 2.52·15-s − 0.622·17-s − 2.30·19-s − 2.67·21-s − 6.63·23-s − 4.10·25-s + 3.04·27-s − 9.44·29-s − 6.80·31-s − 2.67·33-s − 0.943·35-s + 3.75·37-s − 2.67·39-s + 0.0796·41-s + 6.71·43-s + 3.90·45-s + 4.55·47-s + 49-s − 1.66·51-s − 5.69·53-s − 0.943·55-s + ⋯
L(s)  = 1  + 1.54·3-s + 0.422·5-s − 0.377·7-s + 1.38·9-s − 0.301·11-s − 0.277·13-s + 0.651·15-s − 0.150·17-s − 0.527·19-s − 0.583·21-s − 1.38·23-s − 0.821·25-s + 0.586·27-s − 1.75·29-s − 1.22·31-s − 0.465·33-s − 0.159·35-s + 0.616·37-s − 0.427·39-s + 0.0124·41-s + 1.02·43-s + 0.582·45-s + 0.664·47-s + 0.142·49-s − 0.232·51-s − 0.782·53-s − 0.127·55-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 \( 1 - 2.67T + 3T^{2} \)
5 \( 1 - 0.943T + 5T^{2} \)
17 \( 1 + 0.622T + 17T^{2} \)
19 \( 1 + 2.30T + 19T^{2} \)
23 \( 1 + 6.63T + 23T^{2} \)
29 \( 1 + 9.44T + 29T^{2} \)
31 \( 1 + 6.80T + 31T^{2} \)
37 \( 1 - 3.75T + 37T^{2} \)
41 \( 1 - 0.0796T + 41T^{2} \)
43 \( 1 - 6.71T + 43T^{2} \)
47 \( 1 - 4.55T + 47T^{2} \)
53 \( 1 + 5.69T + 53T^{2} \)
59 \( 1 + 5.42T + 59T^{2} \)
61 \( 1 + 5.72T + 61T^{2} \)
67 \( 1 - 0.881T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 9.92T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 5.11T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 + 6.32T + 97T^{2} \)
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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

−7.71586159606257825074479776073, −7.03665510362908070214453790668, −6.03376320036486143310715427901, −5.55955811599540994489374839281, −4.31100363549711321285102742077, −3.84091489917318149411680080699, −3.03246872263447133645241536220, −2.19527184306508229630788350721, −1.77988447269820424813305028915, 0, 1.77988447269820424813305028915, 2.19527184306508229630788350721, 3.03246872263447133645241536220, 3.84091489917318149411680080699, 4.31100363549711321285102742077, 5.55955811599540994489374839281, 6.03376320036486143310715427901, 7.03665510362908070214453790668, 7.71586159606257825074479776073

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