Properties

Label 2-8011-1.1-c1-0-117
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.61·2-s + 1.26·3-s + 4.85·4-s + 0.306·5-s − 3.30·6-s − 1.43·7-s − 7.47·8-s − 1.41·9-s − 0.802·10-s + 2.14·11-s + 6.11·12-s − 3.19·13-s + 3.76·14-s + 0.386·15-s + 9.85·16-s − 3.80·17-s + 3.69·18-s + 4.63·19-s + 1.48·20-s − 1.81·21-s − 5.61·22-s − 3.20·23-s − 9.41·24-s − 4.90·25-s + 8.37·26-s − 5.56·27-s − 6.98·28-s + ⋯
L(s)  = 1  − 1.85·2-s + 0.727·3-s + 2.42·4-s + 0.137·5-s − 1.34·6-s − 0.543·7-s − 2.64·8-s − 0.470·9-s − 0.253·10-s + 0.646·11-s + 1.76·12-s − 0.887·13-s + 1.00·14-s + 0.0997·15-s + 2.46·16-s − 0.923·17-s + 0.870·18-s + 1.06·19-s + 0.332·20-s − 0.395·21-s − 1.19·22-s − 0.668·23-s − 1.92·24-s − 0.981·25-s + 1.64·26-s − 1.07·27-s − 1.31·28-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

\(L(1)\) \(\approx\) \(0.5771997357\)
\(L(\frac12)\) \(\approx\) \(0.5771997357\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.61T + 2T^{2} \)
3 \( 1 - 1.26T + 3T^{2} \)
5 \( 1 - 0.306T + 5T^{2} \)
7 \( 1 + 1.43T + 7T^{2} \)
11 \( 1 - 2.14T + 11T^{2} \)
13 \( 1 + 3.19T + 13T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
19 \( 1 - 4.63T + 19T^{2} \)
23 \( 1 + 3.20T + 23T^{2} \)
29 \( 1 + 3.93T + 29T^{2} \)
31 \( 1 + 3.98T + 31T^{2} \)
37 \( 1 - 3.40T + 37T^{2} \)
41 \( 1 - 8.71T + 41T^{2} \)
43 \( 1 + 7.51T + 43T^{2} \)
47 \( 1 + 8.65T + 47T^{2} \)
53 \( 1 + 1.13T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 1.57T + 61T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 0.359T + 73T^{2} \)
79 \( 1 + 0.646T + 79T^{2} \)
83 \( 1 - 6.65T + 83T^{2} \)
89 \( 1 + 8.54T + 89T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.044209681276593803743468800061, −7.39523340890891233347288630205, −6.77795789414029725958832711226, −6.12335033972654218210746491807, −5.30676319704323973598498505434, −3.92116843585832609539625926280, −3.15082815965666084003107400687, −2.31159966609945927097413968976, −1.79533040451211485753299454613, −0.45500974524589911079864599943, 0.45500974524589911079864599943, 1.79533040451211485753299454613, 2.31159966609945927097413968976, 3.15082815965666084003107400687, 3.92116843585832609539625926280, 5.30676319704323973598498505434, 6.12335033972654218210746491807, 6.77795789414029725958832711226, 7.39523340890891233347288630205, 8.044209681276593803743468800061

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