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  − 2.49·2-s + 4.21·4-s + 0.262·5-s + 7-s − 5.52·8-s − 0.655·10-s − 3.55·11-s − 2.65·13-s − 2.49·14-s + 5.34·16-s − 7.62·17-s − 1.10·19-s + 1.10·20-s + 8.85·22-s + 3.50·23-s − 4.93·25-s + 6.62·26-s + 4.21·28-s + 4.14·29-s + 7.01·31-s − 2.27·32-s + 19.0·34-s + 0.262·35-s − 0.847·37-s + 2.74·38-s − 1.45·40-s + 8.88·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 2.10·4-s + 0.117·5-s + 0.377·7-s − 1.95·8-s − 0.207·10-s − 1.07·11-s − 0.736·13-s − 0.666·14-s + 1.33·16-s − 1.84·17-s − 0.252·19-s + 0.247·20-s + 1.88·22-s + 0.730·23-s − 0.986·25-s + 1.29·26-s + 0.796·28-s + 0.769·29-s + 1.26·31-s − 0.401·32-s + 3.25·34-s + 0.0444·35-s − 0.139·37-s + 0.445·38-s − 0.229·40-s + 1.38·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4088462695$
$L(\frac12)$  $\approx$  $0.4088462695$
$L(\frac{3}{2})$   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 \( 1 + 2.49T + 2T^{2} \)
5 \( 1 - 0.262T + 5T^{2} \)
11 \( 1 + 3.55T + 11T^{2} \)
13 \( 1 + 2.65T + 13T^{2} \)
17 \( 1 + 7.62T + 17T^{2} \)
19 \( 1 + 1.10T + 19T^{2} \)
23 \( 1 - 3.50T + 23T^{2} \)
29 \( 1 - 4.14T + 29T^{2} \)
31 \( 1 - 7.01T + 31T^{2} \)
37 \( 1 + 0.847T + 37T^{2} \)
41 \( 1 - 8.88T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 8.26T + 47T^{2} \)
53 \( 1 + 4.21T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 1.49T + 61T^{2} \)
67 \( 1 + 2.38T + 67T^{2} \)
71 \( 1 - 9.51T + 71T^{2} \)
73 \( 1 + 1.22T + 73T^{2} \)
79 \( 1 - 4.70T + 79T^{2} \)
83 \( 1 + 3.89T + 83T^{2} \)
89 \( 1 + 6.98T + 89T^{2} \)
97 \( 1 + 10.7T + 97T^{2} \)
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

−8.040145103926093603760716727634, −7.38847716150319373535293561400, −6.67629579239750238092644205793, −6.17053731978057713354454240037, −4.99199003129827816738121099646, −4.48435100438885676834301567528, −2.97745282962256382936325933066, −2.36277084962480330552135675518, −1.66129592080275217380740474074, −0.40083653250928281226893572295, 0.40083653250928281226893572295, 1.66129592080275217380740474074, 2.36277084962480330552135675518, 2.97745282962256382936325933066, 4.48435100438885676834301567528, 4.99199003129827816738121099646, 6.17053731978057713354454240037, 6.67629579239750238092644205793, 7.38847716150319373535293561400, 8.040145103926093603760716727634

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