Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 127 $
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  − 0.395·2-s − 1.84·4-s − 1.95·5-s + 7-s + 1.52·8-s + 0.772·10-s + 5.34·11-s + 6.49·13-s − 0.395·14-s + 3.08·16-s − 7.92·17-s − 5.50·19-s + 3.59·20-s − 2.11·22-s + 3.09·23-s − 1.18·25-s − 2.56·26-s − 1.84·28-s + 3.39·29-s + 10.2·31-s − 4.26·32-s + 3.13·34-s − 1.95·35-s + 7.25·37-s + 2.17·38-s − 2.96·40-s + 10.1·41-s + ⋯
L(s)  = 1  − 0.279·2-s − 0.921·4-s − 0.873·5-s + 0.377·7-s + 0.537·8-s + 0.244·10-s + 1.61·11-s + 1.80·13-s − 0.105·14-s + 0.771·16-s − 1.92·17-s − 1.26·19-s + 0.804·20-s − 0.450·22-s + 0.646·23-s − 0.237·25-s − 0.503·26-s − 0.348·28-s + 0.630·29-s + 1.83·31-s − 0.753·32-s + 0.537·34-s − 0.329·35-s + 1.19·37-s + 0.353·38-s − 0.469·40-s + 1.59·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8001\)    =    \(3^{2} \cdot 7 \cdot 127\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8001} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.415749998$
$L(\frac12)$  $\approx$  $1.415749998$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;7,\;127\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;127\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
7 \( 1 - T \)
127 \( 1 - T \)
good2 \( 1 + 0.395T + 2T^{2} \)
5 \( 1 + 1.95T + 5T^{2} \)
11 \( 1 - 5.34T + 11T^{2} \)
13 \( 1 - 6.49T + 13T^{2} \)
17 \( 1 + 7.92T + 17T^{2} \)
19 \( 1 + 5.50T + 19T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 - 3.39T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 7.25T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 - 1.66T + 47T^{2} \)
53 \( 1 + 9.51T + 53T^{2} \)
59 \( 1 - 5.51T + 59T^{2} \)
61 \( 1 - 0.0570T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 + 7.73T + 71T^{2} \)
73 \( 1 - 1.79T + 73T^{2} \)
79 \( 1 - 1.81T + 79T^{2} \)
83 \( 1 + 6.90T + 83T^{2} \)
89 \( 1 + 17.5T + 89T^{2} \)
97 \( 1 - 4.44T + 97T^{2} \)
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

−8.116465760038070950827683694455, −7.22470166012811750262747706492, −6.27435493311400728242451003568, −6.10175363374678746047671575418, −4.56695027237760443302061867592, −4.19830252084406117248903890720, −3.99382864436602056496869112106, −2.71832711837020377185200953095, −1.41767343561810024295191800308, −0.69945726910162737877524927022, 0.69945726910162737877524927022, 1.41767343561810024295191800308, 2.71832711837020377185200953095, 3.99382864436602056496869112106, 4.19830252084406117248903890720, 4.56695027237760443302061867592, 6.10175363374678746047671575418, 6.27435493311400728242451003568, 7.22470166012811750262747706492, 8.116465760038070950827683694455

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