Properties

Label 2-8041-1.1-c1-0-16
Degree $2$
Conductor $8041$
Sign $1$
Analytic cond. $64.2077$
Root an. cond. $8.01297$
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  − 2.05·2-s − 3.10·3-s + 2.22·4-s − 2.37·5-s + 6.38·6-s − 3.40·7-s − 0.468·8-s + 6.64·9-s + 4.89·10-s + 11-s − 6.92·12-s − 5.45·13-s + 6.99·14-s + 7.38·15-s − 3.49·16-s − 17-s − 13.6·18-s + 1.95·19-s − 5.29·20-s + 10.5·21-s − 2.05·22-s − 4.50·23-s + 1.45·24-s + 0.657·25-s + 11.2·26-s − 11.3·27-s − 7.57·28-s + ⋯
L(s)  = 1  − 1.45·2-s − 1.79·3-s + 1.11·4-s − 1.06·5-s + 2.60·6-s − 1.28·7-s − 0.165·8-s + 2.21·9-s + 1.54·10-s + 0.301·11-s − 1.99·12-s − 1.51·13-s + 1.86·14-s + 1.90·15-s − 0.872·16-s − 0.242·17-s − 3.22·18-s + 0.448·19-s − 1.18·20-s + 2.30·21-s − 0.438·22-s − 0.940·23-s + 0.297·24-s + 0.131·25-s + 2.20·26-s − 2.18·27-s − 1.43·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8041\)    =    \(11 \cdot 17 \cdot 43\)
Sign: $1$
Analytic conductor: \(64.2077\)
Root analytic conductor: \(8.01297\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8041,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04459454932\)
\(L(\frac12)\) \(\approx\) \(0.04459454932\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
17 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 + 2.05T + 2T^{2} \)
3 \( 1 + 3.10T + 3T^{2} \)
5 \( 1 + 2.37T + 5T^{2} \)
7 \( 1 + 3.40T + 7T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
19 \( 1 - 1.95T + 19T^{2} \)
23 \( 1 + 4.50T + 23T^{2} \)
29 \( 1 - 5.46T + 29T^{2} \)
31 \( 1 - 6.68T + 31T^{2} \)
37 \( 1 - 1.68T + 37T^{2} \)
41 \( 1 + 8.75T + 41T^{2} \)
47 \( 1 + 3.32T + 47T^{2} \)
53 \( 1 + 3.01T + 53T^{2} \)
59 \( 1 - 0.487T + 59T^{2} \)
61 \( 1 - 6.07T + 61T^{2} \)
67 \( 1 + 0.331T + 67T^{2} \)
71 \( 1 + 0.741T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 - 2.19T + 83T^{2} \)
89 \( 1 + 4.14T + 89T^{2} \)
97 \( 1 + 1.97T + 97T^{2} \)
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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

−7.82405226396913488426529356487, −6.97716699711266281701960787435, −6.76082894692344726732196222475, −6.08524919870296342423193981160, −5.00715644499060504531816150556, −4.49799341023500387726065955006, −3.57368890344658605104732059488, −2.36767973885640042559380837321, −1.06263304215027803160154797817, −0.17765518179813867084385107607, 0.17765518179813867084385107607, 1.06263304215027803160154797817, 2.36767973885640042559380837321, 3.57368890344658605104732059488, 4.49799341023500387726065955006, 5.00715644499060504531816150556, 6.08524919870296342423193981160, 6.76082894692344726732196222475, 6.97716699711266281701960787435, 7.82405226396913488426529356487

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