Properties

Label 2-8011-1.1-c1-0-83
Degree $2$
Conductor $8011$
Sign $1$
Analytic cond. $63.9681$
Root an. cond. $7.99800$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·2-s − 2.88·3-s + 5.34·4-s − 1.42·5-s + 7.83·6-s − 3.83·7-s − 9.05·8-s + 5.34·9-s + 3.87·10-s − 1.77·11-s − 15.4·12-s − 2.78·13-s + 10.3·14-s + 4.12·15-s + 13.8·16-s + 5.22·17-s − 14.4·18-s + 1.32·19-s − 7.63·20-s + 11.0·21-s + 4.80·22-s − 7.98·23-s + 26.1·24-s − 2.95·25-s + 7.54·26-s − 6.78·27-s − 20.4·28-s + ⋯
L(s)  = 1  − 1.91·2-s − 1.66·3-s + 2.67·4-s − 0.638·5-s + 3.19·6-s − 1.44·7-s − 3.20·8-s + 1.78·9-s + 1.22·10-s − 0.534·11-s − 4.45·12-s − 0.772·13-s + 2.77·14-s + 1.06·15-s + 3.46·16-s + 1.26·17-s − 3.41·18-s + 0.303·19-s − 1.70·20-s + 2.41·21-s + 1.02·22-s − 1.66·23-s + 5.34·24-s − 0.591·25-s + 1.47·26-s − 1.30·27-s − 3.87·28-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

Degree: \(2\)
Conductor: \(8011\)
Sign: $1$
Analytic conductor: \(63.9681\)
Root analytic conductor: \(7.99800\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8011,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1542781111\)
\(L(\frac12)\) \(\approx\) \(0.1542781111\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad8011 \( 1+O(T) \)
good2 \( 1 + 2.70T + 2T^{2} \)
3 \( 1 + 2.88T + 3T^{2} \)
5 \( 1 + 1.42T + 5T^{2} \)
7 \( 1 + 3.83T + 7T^{2} \)
11 \( 1 + 1.77T + 11T^{2} \)
13 \( 1 + 2.78T + 13T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + 7.98T + 23T^{2} \)
29 \( 1 - 8.30T + 29T^{2} \)
31 \( 1 - 4.90T + 31T^{2} \)
37 \( 1 - 9.12T + 37T^{2} \)
41 \( 1 - 11.9T + 41T^{2} \)
43 \( 1 + 7.89T + 43T^{2} \)
47 \( 1 + 3.43T + 47T^{2} \)
53 \( 1 - 5.91T + 53T^{2} \)
59 \( 1 + 0.635T + 59T^{2} \)
61 \( 1 + 3.47T + 61T^{2} \)
67 \( 1 + 3.67T + 67T^{2} \)
71 \( 1 - 2.70T + 71T^{2} \)
73 \( 1 + 2.67T + 73T^{2} \)
79 \( 1 - 9.08T + 79T^{2} \)
83 \( 1 - 9.99T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.73282290453000120451710915243, −7.34709261283220644848194737078, −6.38682547473726402314589594697, −6.22240946352607067550758016419, −5.47412525475847397464199021306, −4.33295301665560279926312217078, −3.21658919359278242308789093800, −2.41125918514363925204713296531, −1.02496770993299840523031749147, −0.35806568112831599652756511450, 0.35806568112831599652756511450, 1.02496770993299840523031749147, 2.41125918514363925204713296531, 3.21658919359278242308789093800, 4.33295301665560279926312217078, 5.47412525475847397464199021306, 6.22240946352607067550758016419, 6.38682547473726402314589594697, 7.34709261283220644848194737078, 7.73282290453000120451710915243

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