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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.93·3-s + 2.25·5-s − 7-s + 0.745·9-s − 11-s + 13-s − 4.36·15-s + 0.616·17-s + 2.42·19-s + 1.93·21-s − 7.69·23-s + 0.0809·25-s + 4.36·27-s + 8.37·29-s − 3.31·31-s + 1.93·33-s − 2.25·35-s + 4.68·37-s − 1.93·39-s − 12.2·41-s + 2.23·43-s + 1.68·45-s − 4.85·47-s + 49-s − 1.19·51-s + 6.99·53-s − 2.25·55-s + ⋯
L(s)  = 1  − 1.11·3-s + 1.00·5-s − 0.377·7-s + 0.248·9-s − 0.301·11-s + 0.277·13-s − 1.12·15-s + 0.149·17-s + 0.556·19-s + 0.422·21-s − 1.60·23-s + 0.0161·25-s + 0.839·27-s + 1.55·29-s − 0.596·31-s + 0.336·33-s − 0.381·35-s + 0.769·37-s − 0.309·39-s − 1.90·41-s + 0.341·43-s + 0.250·45-s − 0.708·47-s + 0.142·49-s − 0.167·51-s + 0.960·53-s − 0.303·55-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 \( 1 + 1.93T + 3T^{2} \)
5 \( 1 - 2.25T + 5T^{2} \)
17 \( 1 - 0.616T + 17T^{2} \)
19 \( 1 - 2.42T + 19T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 - 8.37T + 29T^{2} \)
31 \( 1 + 3.31T + 31T^{2} \)
37 \( 1 - 4.68T + 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 2.23T + 43T^{2} \)
47 \( 1 + 4.85T + 47T^{2} \)
53 \( 1 - 6.99T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 2.74T + 61T^{2} \)
67 \( 1 + 5.42T + 67T^{2} \)
71 \( 1 + 2.38T + 71T^{2} \)
73 \( 1 + 4.98T + 73T^{2} \)
79 \( 1 + 9.31T + 79T^{2} \)
83 \( 1 + 3.95T + 83T^{2} \)
89 \( 1 - 6.91T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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

−7.29932741492675772219353689073, −6.50955220488829500330168221975, −6.06734875968744111386927469730, −5.53078847487313590767644337373, −4.95101429408696187818060667979, −4.01280813357970430282356885122, −3.02755822789317687495521529885, −2.14781565556594649594445845737, −1.14821143967501402151942184797, 0, 1.14821143967501402151942184797, 2.14781565556594649594445845737, 3.02755822789317687495521529885, 4.01280813357970430282356885122, 4.95101429408696187818060667979, 5.53078847487313590767644337373, 6.06734875968744111386927469730, 6.50955220488829500330168221975, 7.29932741492675772219353689073

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