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  − 1.21·3-s + 2.90·5-s − 7-s − 1.51·9-s + 11-s − 13-s − 3.53·15-s + 3.20·17-s − 0.338·19-s + 1.21·21-s + 7.17·23-s + 3.41·25-s + 5.50·27-s + 10.5·29-s + 1.34·31-s − 1.21·33-s − 2.90·35-s + 2.97·37-s + 1.21·39-s − 3.83·41-s − 2.40·43-s − 4.39·45-s − 12.1·47-s + 49-s − 3.91·51-s + 1.77·53-s + 2.90·55-s + ⋯
L(s)  = 1  − 0.703·3-s + 1.29·5-s − 0.377·7-s − 0.505·9-s + 0.301·11-s − 0.277·13-s − 0.912·15-s + 0.778·17-s − 0.0775·19-s + 0.265·21-s + 1.49·23-s + 0.682·25-s + 1.05·27-s + 1.96·29-s + 0.241·31-s − 0.212·33-s − 0.490·35-s + 0.489·37-s + 0.195·39-s − 0.598·41-s − 0.367·43-s − 0.655·45-s − 1.76·47-s + 0.142·49-s − 0.547·51-s + 0.243·53-s + 0.391·55-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  =  $(2,\ 8008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.938181005$
$L(\frac12)$  $\approx$  $1.938181005$
$L(\frac{3}{2})$   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 \( 1 + 1.21T + 3T^{2} \)
5 \( 1 - 2.90T + 5T^{2} \)
17 \( 1 - 3.20T + 17T^{2} \)
19 \( 1 + 0.338T + 19T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 - 10.5T + 29T^{2} \)
31 \( 1 - 1.34T + 31T^{2} \)
37 \( 1 - 2.97T + 37T^{2} \)
41 \( 1 + 3.83T + 41T^{2} \)
43 \( 1 + 2.40T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 - 1.77T + 53T^{2} \)
59 \( 1 + 6.34T + 59T^{2} \)
61 \( 1 + 6.55T + 61T^{2} \)
67 \( 1 + 0.479T + 67T^{2} \)
71 \( 1 + 7.64T + 71T^{2} \)
73 \( 1 + 5.07T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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

−7.82206142010795169486216383887, −6.78086546101236746914593669953, −6.40374895780874632979625569492, −5.84113055144929514191043817224, −5.08451317282772658023883190520, −4.66538881875752281082322894145, −3.22219992888333645578336719539, −2.79092782569540392730747516631, −1.64518336704685956304102483762, −0.73890267027712775087779390446, 0.73890267027712775087779390446, 1.64518336704685956304102483762, 2.79092782569540392730747516631, 3.22219992888333645578336719539, 4.66538881875752281082322894145, 5.08451317282772658023883190520, 5.84113055144929514191043817224, 6.40374895780874632979625569492, 6.78086546101236746914593669953, 7.82206142010795169486216383887

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