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  − 1.36·3-s − 3.92·5-s + 7-s − 1.12·9-s − 11-s − 13-s + 5.37·15-s − 2.95·17-s − 3.52·19-s − 1.36·21-s + 4.25·23-s + 10.3·25-s + 5.64·27-s − 9.21·29-s + 1.84·31-s + 1.36·33-s − 3.92·35-s + 0.471·37-s + 1.36·39-s + 6.70·41-s + 12.6·43-s + 4.41·45-s + 5.58·47-s + 49-s + 4.05·51-s + 12.0·53-s + 3.92·55-s + ⋯
L(s)  = 1  − 0.790·3-s − 1.75·5-s + 0.377·7-s − 0.375·9-s − 0.301·11-s − 0.277·13-s + 1.38·15-s − 0.717·17-s − 0.807·19-s − 0.298·21-s + 0.886·23-s + 2.07·25-s + 1.08·27-s − 1.71·29-s + 0.332·31-s + 0.238·33-s − 0.663·35-s + 0.0775·37-s + 0.219·39-s + 1.04·41-s + 1.92·43-s + 0.658·45-s + 0.814·47-s + 0.142·49-s + 0.567·51-s + 1.66·53-s + 0.529·55-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 \( 1 + 1.36T + 3T^{2} \)
5 \( 1 + 3.92T + 5T^{2} \)
17 \( 1 + 2.95T + 17T^{2} \)
19 \( 1 + 3.52T + 19T^{2} \)
23 \( 1 - 4.25T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 - 0.471T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 - 5.58T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 7.88T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 + 0.782T + 67T^{2} \)
71 \( 1 + 0.831T + 71T^{2} \)
73 \( 1 + 7.35T + 73T^{2} \)
79 \( 1 - 3.31T + 79T^{2} \)
83 \( 1 - 12.9T + 83T^{2} \)
89 \( 1 + 9.73T + 89T^{2} \)
97 \( 1 + 0.699T + 97T^{2} \)
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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

−7.49569837956531194722615215422, −6.96182303945641722583459679256, −6.03332955466003676439600478493, −5.38960126743585175673723370337, −4.46127911946999236177084813916, −4.21847233972453388057733785509, −3.17597622235406965217435100405, −2.33222925126828449600265891159, −0.824736239424921146838993793356, 0, 0.824736239424921146838993793356, 2.33222925126828449600265891159, 3.17597622235406965217435100405, 4.21847233972453388057733785509, 4.46127911946999236177084813916, 5.38960126743585175673723370337, 6.03332955466003676439600478493, 6.96182303945641722583459679256, 7.49569837956531194722615215422

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