Properties

Label 2-8022-1.1-c1-0-100
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 + 4.12·5-s − 6-s + 7-s − 8-s + 9-s − 4.12·10-s − 0.209·11-s + 12-s + 2.02·13-s − 14-s + 4.12·15-s + 16-s − 6.22·17-s − 18-s + 7.05·19-s + 4.12·20-s + 21-s + 0.209·22-s + 4.78·23-s − 24-s + 12.0·25-s − 2.02·26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.84·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 1.30·10-s − 0.0631·11-s + 0.288·12-s + 0.562·13-s − 0.267·14-s + 1.06·15-s + 0.250·16-s − 1.51·17-s − 0.235·18-s + 1.61·19-s + 0.923·20-s + 0.218·21-s + 0.0446·22-s + 0.997·23-s − 0.204·24-s + 2.40·25-s − 0.397·26-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

\(L(1)\) \(\approx\) \(3.212408518\)
\(L(\frac12)\) \(\approx\) \(3.212408518\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.12T + 5T^{2} \)
11 \( 1 + 0.209T + 11T^{2} \)
13 \( 1 - 2.02T + 13T^{2} \)
17 \( 1 + 6.22T + 17T^{2} \)
19 \( 1 - 7.05T + 19T^{2} \)
23 \( 1 - 4.78T + 23T^{2} \)
29 \( 1 + 6.72T + 29T^{2} \)
31 \( 1 - 3.59T + 31T^{2} \)
37 \( 1 + 4.94T + 37T^{2} \)
41 \( 1 - 3.48T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 2.83T + 47T^{2} \)
53 \( 1 - 0.180T + 53T^{2} \)
59 \( 1 + 1.70T + 59T^{2} \)
61 \( 1 + 5.32T + 61T^{2} \)
67 \( 1 + 2.99T + 67T^{2} \)
71 \( 1 + 5.25T + 71T^{2} \)
73 \( 1 + 1.40T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 2.56T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−7.88763801398963217263220880097, −7.15381774618137305441961037895, −6.58593415168935624794610927570, −5.77739466289550756963152971871, −5.27360503651503644386051445682, −4.32100341154554580165518222859, −3.12000080647599501635546157802, −2.47447202450941407588078984856, −1.73875114708461622324652208907, −1.03826005146978867216371929103, 1.03826005146978867216371929103, 1.73875114708461622324652208907, 2.47447202450941407588078984856, 3.12000080647599501635546157802, 4.32100341154554580165518222859, 5.27360503651503644386051445682, 5.77739466289550756963152971871, 6.58593415168935624794610927570, 7.15381774618137305441961037895, 7.88763801398963217263220880097

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