Properties

Degree 2
Conductor 8011
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  − 2.59·2-s − 2.89·3-s + 4.74·4-s + 2.71·5-s + 7.51·6-s − 3.02·7-s − 7.12·8-s + 5.37·9-s − 7.05·10-s + 6.06·11-s − 13.7·12-s − 0.0981·13-s + 7.85·14-s − 7.86·15-s + 9.02·16-s + 1.66·17-s − 13.9·18-s + 1.73·19-s + 12.8·20-s + 8.75·21-s − 15.7·22-s + 0.496·23-s + 20.6·24-s + 2.37·25-s + 0.254·26-s − 6.88·27-s − 14.3·28-s + ⋯
L(s)  = 1  − 1.83·2-s − 1.67·3-s + 2.37·4-s + 1.21·5-s + 3.06·6-s − 1.14·7-s − 2.52·8-s + 1.79·9-s − 2.23·10-s + 1.82·11-s − 3.96·12-s − 0.0272·13-s + 2.10·14-s − 2.02·15-s + 2.25·16-s + 0.404·17-s − 3.29·18-s + 0.396·19-s + 2.88·20-s + 1.91·21-s − 3.36·22-s + 0.103·23-s + 4.21·24-s + 0.475·25-s + 0.0499·26-s − 1.32·27-s − 2.71·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8011\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8011} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8011,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6262216654$
$L(\frac12)$  $\approx$  $0.6262216654$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 8011$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p = 8011$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad8011 \( 1+O(T) \)
good2 \( 1 + 2.59T + 2T^{2} \)
3 \( 1 + 2.89T + 3T^{2} \)
5 \( 1 - 2.71T + 5T^{2} \)
7 \( 1 + 3.02T + 7T^{2} \)
11 \( 1 - 6.06T + 11T^{2} \)
13 \( 1 + 0.0981T + 13T^{2} \)
17 \( 1 - 1.66T + 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 - 0.496T + 23T^{2} \)
29 \( 1 + 3.28T + 29T^{2} \)
31 \( 1 - 4.68T + 31T^{2} \)
37 \( 1 - 8.03T + 37T^{2} \)
41 \( 1 - 1.40T + 41T^{2} \)
43 \( 1 - 9.32T + 43T^{2} \)
47 \( 1 + 4.83T + 47T^{2} \)
53 \( 1 + 8.62T + 53T^{2} \)
59 \( 1 - 10.5T + 59T^{2} \)
61 \( 1 - 9.06T + 61T^{2} \)
67 \( 1 + 9.24T + 67T^{2} \)
71 \( 1 + 5.78T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 7.18T + 79T^{2} \)
83 \( 1 + 5.11T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 6.33T + 97T^{2} \)
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

−7.72049332071424576461461321540, −6.94626293909445829830322985547, −6.48226539576128360944975145940, −6.09210754839792822959231373465, −5.63802966421705411126106336297, −4.40171556043004106304142149861, −3.25552893167317500438937365185, −2.11561607321168112116287254422, −1.22096180429786270773658529789, −0.67215859857247124167826788842, 0.67215859857247124167826788842, 1.22096180429786270773658529789, 2.11561607321168112116287254422, 3.25552893167317500438937365185, 4.40171556043004106304142149861, 5.63802966421705411126106336297, 6.09210754839792822959231373465, 6.48226539576128360944975145940, 6.94626293909445829830322985547, 7.72049332071424576461461321540

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