Properties

Label 2-8041-1.1-c1-0-80
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  − 2.22·2-s − 3.23·3-s + 2.93·4-s − 1.33·5-s + 7.19·6-s − 0.932·7-s − 2.06·8-s + 7.48·9-s + 2.95·10-s + 11-s − 9.49·12-s + 4.80·13-s + 2.07·14-s + 4.31·15-s − 1.27·16-s + 17-s − 16.6·18-s + 0.807·19-s − 3.90·20-s + 3.01·21-s − 2.22·22-s − 8.62·23-s + 6.69·24-s − 3.22·25-s − 10.6·26-s − 14.5·27-s − 2.73·28-s + ⋯
L(s)  = 1  − 1.57·2-s − 1.86·3-s + 1.46·4-s − 0.595·5-s + 2.93·6-s − 0.352·7-s − 0.730·8-s + 2.49·9-s + 0.934·10-s + 0.301·11-s − 2.74·12-s + 1.33·13-s + 0.553·14-s + 1.11·15-s − 0.317·16-s + 0.242·17-s − 3.92·18-s + 0.185·19-s − 0.872·20-s + 0.658·21-s − 0.473·22-s − 1.79·23-s + 1.36·24-s − 0.645·25-s − 2.09·26-s − 2.79·27-s − 0.516·28-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

\(L(1)\) \(\approx\) \(0.2804945036\)
\(L(\frac12)\) \(\approx\) \(0.2804945036\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.22T + 2T^{2} \)
3 \( 1 + 3.23T + 3T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
7 \( 1 + 0.932T + 7T^{2} \)
13 \( 1 - 4.80T + 13T^{2} \)
19 \( 1 - 0.807T + 19T^{2} \)
23 \( 1 + 8.62T + 23T^{2} \)
29 \( 1 + 7.30T + 29T^{2} \)
31 \( 1 - 8.01T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 - 8.36T + 41T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 + 0.149T + 61T^{2} \)
67 \( 1 + 9.07T + 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 - 2.14T + 79T^{2} \)
83 \( 1 - 8.57T + 83T^{2} \)
89 \( 1 + 8.59T + 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−7.83845281005894374469854676882, −7.15866807833945171617746341290, −6.54894257360345504396032386642, −5.96375201992037900879277274171, −5.39430423807893619429265521672, −4.13042621092407835923810269827, −3.83011588483539691511688229202, −2.08941140610068219895152253812, −1.17798207000594203089690592035, −0.44109729313636911525899981725, 0.44109729313636911525899981725, 1.17798207000594203089690592035, 2.08941140610068219895152253812, 3.83011588483539691511688229202, 4.13042621092407835923810269827, 5.39430423807893619429265521672, 5.96375201992037900879277274171, 6.54894257360345504396032386642, 7.15866807833945171617746341290, 7.83845281005894374469854676882

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