Properties

Label 2-8022-1.1-c1-0-92
Degree $2$
Conductor $8022$
Sign $1$
Analytic cond. $64.0559$
Root an. cond. $8.00349$
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-s + 3-s + 4-s − 2.97·5-s + 6-s + 7-s + 8-s + 9-s − 2.97·10-s + 4.71·11-s + 12-s + 4.48·13-s + 14-s − 2.97·15-s + 16-s + 1.10·17-s + 18-s + 5.53·19-s − 2.97·20-s + 21-s + 4.71·22-s + 0.605·23-s + 24-s + 3.87·25-s + 4.48·26-s + 27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.33·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.941·10-s + 1.42·11-s + 0.288·12-s + 1.24·13-s + 0.267·14-s − 0.769·15-s + 0.250·16-s + 0.268·17-s + 0.235·18-s + 1.27·19-s − 0.665·20-s + 0.218·21-s + 1.00·22-s + 0.126·23-s + 0.204·24-s + 0.774·25-s + 0.879·26-s + 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8022\)    =    \(2 \cdot 3 \cdot 7 \cdot 191\)
Sign: $1$
Analytic conductor: \(64.0559\)
Root analytic conductor: \(8.00349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8022,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.315006302\)
\(L(\frac12)\) \(\approx\) \(4.315006302\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 - T \)
191 \( 1 + T \)
good5 \( 1 + 2.97T + 5T^{2} \)
11 \( 1 - 4.71T + 11T^{2} \)
13 \( 1 - 4.48T + 13T^{2} \)
17 \( 1 - 1.10T + 17T^{2} \)
19 \( 1 - 5.53T + 19T^{2} \)
23 \( 1 - 0.605T + 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 - 3.25T + 37T^{2} \)
41 \( 1 - 8.59T + 41T^{2} \)
43 \( 1 + 3.30T + 43T^{2} \)
47 \( 1 + 9.81T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 2.94T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + 2.14T + 67T^{2} \)
71 \( 1 + 3.79T + 71T^{2} \)
73 \( 1 + 6.45T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 6.68T + 97T^{2} \)
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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

−7.79713649415974547041067180972, −7.24966807497697579447479993995, −6.43665138456470497300015429105, −5.80823416775685332783637042388, −4.69058095554482000414257305801, −4.23235930768759382584825520185, −3.40272142528162539969289236492, −3.24285584105587324954254493432, −1.73594671974393184558687890628, −0.983211542402108402158440735133, 0.983211542402108402158440735133, 1.73594671974393184558687890628, 3.24285584105587324954254493432, 3.40272142528162539969289236492, 4.23235930768759382584825520185, 4.69058095554482000414257305801, 5.80823416775685332783637042388, 6.43665138456470497300015429105, 7.24966807497697579447479993995, 7.79713649415974547041067180972

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