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  + 0.141·3-s + 0.343·5-s − 7-s − 2.97·9-s + 11-s + 13-s + 0.0486·15-s − 1.94·17-s − 0.383·19-s − 0.141·21-s − 1.66·23-s − 4.88·25-s − 0.846·27-s + 3.52·29-s + 8.92·31-s + 0.141·33-s − 0.343·35-s + 5.87·37-s + 0.141·39-s + 6.78·41-s − 1.87·43-s − 1.02·45-s + 2.51·47-s + 49-s − 0.275·51-s + 0.793·53-s + 0.343·55-s + ⋯
L(s)  = 1  + 0.0817·3-s + 0.153·5-s − 0.377·7-s − 0.993·9-s + 0.301·11-s + 0.277·13-s + 0.0125·15-s − 0.471·17-s − 0.0880·19-s − 0.0308·21-s − 0.347·23-s − 0.976·25-s − 0.162·27-s + 0.655·29-s + 1.60·31-s + 0.0246·33-s − 0.0581·35-s + 0.965·37-s + 0.0226·39-s + 1.05·41-s − 0.286·43-s − 0.152·45-s + 0.366·47-s + 0.142·49-s − 0.0385·51-s + 0.108·53-s + 0.0463·55-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 \( 1 - 0.141T + 3T^{2} \)
5 \( 1 - 0.343T + 5T^{2} \)
17 \( 1 + 1.94T + 17T^{2} \)
19 \( 1 + 0.383T + 19T^{2} \)
23 \( 1 + 1.66T + 23T^{2} \)
29 \( 1 - 3.52T + 29T^{2} \)
31 \( 1 - 8.92T + 31T^{2} \)
37 \( 1 - 5.87T + 37T^{2} \)
41 \( 1 - 6.78T + 41T^{2} \)
43 \( 1 + 1.87T + 43T^{2} \)
47 \( 1 - 2.51T + 47T^{2} \)
53 \( 1 - 0.793T + 53T^{2} \)
59 \( 1 + 14.0T + 59T^{2} \)
61 \( 1 - 6.36T + 61T^{2} \)
67 \( 1 + 8.05T + 67T^{2} \)
71 \( 1 - 0.428T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 9.37T + 79T^{2} \)
83 \( 1 + 4.37T + 83T^{2} \)
89 \( 1 + 4.95T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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

−7.61850407614653280347100567089, −6.58103652266934672820967762012, −6.16433392930943096906685808939, −5.57413645180659260664473490946, −4.55882770411562545675219876313, −3.95510625694091823356048027662, −2.94321831668259577596440716076, −2.43553338247583224016173300875, −1.22335425551692589712406716149, 0, 1.22335425551692589712406716149, 2.43553338247583224016173300875, 2.94321831668259577596440716076, 3.95510625694091823356048027662, 4.55882770411562545675219876313, 5.57413645180659260664473490946, 6.16433392930943096906685808939, 6.58103652266934672820967762012, 7.61850407614653280347100567089

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