Properties

Degree 2
Conductor $ 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.63·3-s − 1.01·5-s − 7-s − 0.341·9-s + 11-s − 13-s + 1.64·15-s + 6.58·17-s − 7.83·19-s + 1.63·21-s − 8.66·23-s − 3.97·25-s + 5.44·27-s − 2.21·29-s + 2.76·31-s − 1.63·33-s + 1.01·35-s + 0.284·37-s + 1.63·39-s − 4.59·41-s + 7.84·43-s + 0.345·45-s − 7.89·47-s + 49-s − 10.7·51-s − 2.84·53-s − 1.01·55-s + ⋯
L(s)  = 1  − 0.941·3-s − 0.451·5-s − 0.377·7-s − 0.113·9-s + 0.301·11-s − 0.277·13-s + 0.425·15-s + 1.59·17-s − 1.79·19-s + 0.355·21-s − 1.80·23-s − 0.795·25-s + 1.04·27-s − 0.412·29-s + 0.497·31-s − 0.283·33-s + 0.170·35-s + 0.0467·37-s + 0.261·39-s − 0.717·41-s + 1.19·43-s + 0.0514·45-s − 1.15·47-s + 0.142·49-s − 1.50·51-s − 0.391·53-s − 0.136·55-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  =  0
Selberg data  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4864144281$
$L(\frac12)$  $\approx$  $0.4864144281$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 + 1.63T + 3T^{2} \)
5 \( 1 + 1.01T + 5T^{2} \)
17 \( 1 - 6.58T + 17T^{2} \)
19 \( 1 + 7.83T + 19T^{2} \)
23 \( 1 + 8.66T + 23T^{2} \)
29 \( 1 + 2.21T + 29T^{2} \)
31 \( 1 - 2.76T + 31T^{2} \)
37 \( 1 - 0.284T + 37T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 - 7.84T + 43T^{2} \)
47 \( 1 + 7.89T + 47T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 + 0.589T + 59T^{2} \)
61 \( 1 - 9.66T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 - 7.88T + 71T^{2} \)
73 \( 1 + 6.45T + 73T^{2} \)
79 \( 1 + 5.04T + 79T^{2} \)
83 \( 1 - 0.656T + 83T^{2} \)
89 \( 1 - 8.01T + 89T^{2} \)
97 \( 1 + 5.20T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.945191201025070405415842038312, −7.03736988938916074645029198377, −6.21084767129638508474991206285, −5.94593868449618196810003284163, −5.14659515062559066741346505157, −4.23524563299615765586709843093, −3.71355644472087826741925407948, −2.69174868277475657021902573956, −1.65173006740566129486036786416, −0.35703921593915835108244291141, 0.35703921593915835108244291141, 1.65173006740566129486036786416, 2.69174868277475657021902573956, 3.71355644472087826741925407948, 4.23524563299615765586709843093, 5.14659515062559066741346505157, 5.94593868449618196810003284163, 6.21084767129638508474991206285, 7.03736988938916074645029198377, 7.945191201025070405415842038312

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