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  + 2.46·3-s − 1.12·5-s + 7-s + 3.05·9-s − 11-s − 13-s − 2.77·15-s + 6.31·17-s − 1.83·19-s + 2.46·21-s − 8.30·23-s − 3.72·25-s + 0.139·27-s − 7.64·29-s − 5.82·31-s − 2.46·33-s − 1.12·35-s + 4.38·37-s − 2.46·39-s − 1.61·41-s − 9.78·43-s − 3.44·45-s + 3.50·47-s + 49-s + 15.5·51-s + 10.6·53-s + 1.12·55-s + ⋯
L(s)  = 1  + 1.42·3-s − 0.504·5-s + 0.377·7-s + 1.01·9-s − 0.301·11-s − 0.277·13-s − 0.717·15-s + 1.53·17-s − 0.420·19-s + 0.537·21-s − 1.73·23-s − 0.745·25-s + 0.0268·27-s − 1.42·29-s − 1.04·31-s − 0.428·33-s − 0.190·35-s + 0.721·37-s − 0.394·39-s − 0.251·41-s − 1.49·43-s − 0.514·45-s + 0.510·47-s + 0.142·49-s + 2.17·51-s + 1.46·53-s + 0.152·55-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 \( 1 - 2.46T + 3T^{2} \)
5 \( 1 + 1.12T + 5T^{2} \)
17 \( 1 - 6.31T + 17T^{2} \)
19 \( 1 + 1.83T + 19T^{2} \)
23 \( 1 + 8.30T + 23T^{2} \)
29 \( 1 + 7.64T + 29T^{2} \)
31 \( 1 + 5.82T + 31T^{2} \)
37 \( 1 - 4.38T + 37T^{2} \)
41 \( 1 + 1.61T + 41T^{2} \)
43 \( 1 + 9.78T + 43T^{2} \)
47 \( 1 - 3.50T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 - 7.27T + 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 8.87T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 - 5.76T + 89T^{2} \)
97 \( 1 + 7.88T + 97T^{2} \)
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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

−7.46637902570839904548389531002, −7.39844675710195879697804049141, −5.96998906352761119815099785006, −5.47424624731945426026042569783, −4.34535566046405077398806634428, −3.77139960419112355453098144492, −3.20363754886686682504820077493, −2.22471385760089905366997795149, −1.62300830559309975651488187097, 0, 1.62300830559309975651488187097, 2.22471385760089905366997795149, 3.20363754886686682504820077493, 3.77139960419112355453098144492, 4.34535566046405077398806634428, 5.47424624731945426026042569783, 5.96998906352761119815099785006, 7.39844675710195879697804049141, 7.46637902570839904548389531002

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