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  − 1.88·2-s + 1.56·4-s + 3.95·5-s + 7-s + 0.823·8-s − 7.47·10-s − 2.14·11-s + 4.63·13-s − 1.88·14-s − 4.68·16-s + 2.02·17-s + 5.69·19-s + 6.18·20-s + 4.05·22-s + 1.45·23-s + 10.6·25-s − 8.75·26-s + 1.56·28-s + 7.67·29-s + 9.00·31-s + 7.19·32-s − 3.83·34-s + 3.95·35-s + 7.50·37-s − 10.7·38-s + 3.25·40-s − 1.70·41-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.781·4-s + 1.76·5-s + 0.377·7-s + 0.291·8-s − 2.36·10-s − 0.647·11-s + 1.28·13-s − 0.504·14-s − 1.17·16-s + 0.492·17-s + 1.30·19-s + 1.38·20-s + 0.864·22-s + 0.303·23-s + 2.13·25-s − 1.71·26-s + 0.295·28-s + 1.42·29-s + 1.61·31-s + 1.27·32-s − 0.657·34-s + 0.668·35-s + 1.23·37-s − 1.74·38-s + 0.515·40-s − 0.266·41-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  =  $(2,\ 8001,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.966096373$
$L(\frac12)$  $\approx$  $1.966096373$
$L(\frac{3}{2})$   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 \( 1 + 1.88T + 2T^{2} \)
5 \( 1 - 3.95T + 5T^{2} \)
11 \( 1 + 2.14T + 11T^{2} \)
13 \( 1 - 4.63T + 13T^{2} \)
17 \( 1 - 2.02T + 17T^{2} \)
19 \( 1 - 5.69T + 19T^{2} \)
23 \( 1 - 1.45T + 23T^{2} \)
29 \( 1 - 7.67T + 29T^{2} \)
31 \( 1 - 9.00T + 31T^{2} \)
37 \( 1 - 7.50T + 37T^{2} \)
41 \( 1 + 1.70T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 + 7.66T + 53T^{2} \)
59 \( 1 - 3.85T + 59T^{2} \)
61 \( 1 + 7.14T + 61T^{2} \)
67 \( 1 + 5.67T + 67T^{2} \)
71 \( 1 - 6.26T + 71T^{2} \)
73 \( 1 + 9.32T + 73T^{2} \)
79 \( 1 - 8.47T + 79T^{2} \)
83 \( 1 + 6.28T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 1.36T + 97T^{2} \)
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

−8.066645084164328118771542999436, −7.34165338633786308032117390688, −6.40187972523320523446951044300, −6.01299172901429114400631197972, −5.14267720223349393164920031512, −4.52984622906206680010838677446, −3.06103623211918081225630259832, −2.43998465084683870389775417887, −1.29895526245423289068351596336, −1.06910583979085332353640081892, 1.06910583979085332353640081892, 1.29895526245423289068351596336, 2.43998465084683870389775417887, 3.06103623211918081225630259832, 4.52984622906206680010838677446, 5.14267720223349393164920031512, 6.01299172901429114400631197972, 6.40187972523320523446951044300, 7.34165338633786308032117390688, 8.066645084164328118771542999436

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