Properties

Label 2-4022-1.1-c1-0-18
Degree $2$
Conductor $4022$
Sign $1$
Analytic cond. $32.1158$
Root an. cond. $5.66708$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3.08·3-s + 4-s − 0.00834·5-s + 3.08·6-s − 0.451·7-s − 8-s + 6.48·9-s + 0.00834·10-s − 3.57·11-s − 3.08·12-s + 3.37·13-s + 0.451·14-s + 0.0256·15-s + 16-s + 1.64·17-s − 6.48·18-s + 1.82·19-s − 0.00834·20-s + 1.39·21-s + 3.57·22-s + 3.38·23-s + 3.08·24-s − 4.99·25-s − 3.37·26-s − 10.7·27-s − 0.451·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.77·3-s + 0.5·4-s − 0.00373·5-s + 1.25·6-s − 0.170·7-s − 0.353·8-s + 2.16·9-s + 0.00263·10-s − 1.07·11-s − 0.889·12-s + 0.937·13-s + 0.120·14-s + 0.00663·15-s + 0.250·16-s + 0.397·17-s − 1.52·18-s + 0.419·19-s − 0.00186·20-s + 0.303·21-s + 0.762·22-s + 0.706·23-s + 0.628·24-s − 0.999·25-s − 0.662·26-s − 2.06·27-s − 0.0852·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4022\)    =    \(2 \cdot 2011\)
Sign: $1$
Analytic conductor: \(32.1158\)
Root analytic conductor: \(5.66708\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4022,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5615206845\)
\(L(\frac12)\) \(\approx\) \(0.5615206845\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
2011 \( 1 - T \)
good3 \( 1 + 3.08T + 3T^{2} \)
5 \( 1 + 0.00834T + 5T^{2} \)
7 \( 1 + 0.451T + 7T^{2} \)
11 \( 1 + 3.57T + 11T^{2} \)
13 \( 1 - 3.37T + 13T^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 - 3.38T + 23T^{2} \)
29 \( 1 - 0.201T + 29T^{2} \)
31 \( 1 - 1.23T + 31T^{2} \)
37 \( 1 - 7.03T + 37T^{2} \)
41 \( 1 - 8.74T + 41T^{2} \)
43 \( 1 + 2.11T + 43T^{2} \)
47 \( 1 + 5.77T + 47T^{2} \)
53 \( 1 - 8.47T + 53T^{2} \)
59 \( 1 - 4.23T + 59T^{2} \)
61 \( 1 + 2.00T + 61T^{2} \)
67 \( 1 - 2.53T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 2.09T + 73T^{2} \)
79 \( 1 - 3.24T + 79T^{2} \)
83 \( 1 + 2.60T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.308213556642795604391394662977, −7.63138992022474016947705475199, −6.94953239261396067519141721208, −6.10856795548362819651765026537, −5.70777812953831334605653559952, −4.95274792428955110310719918564, −4.00130678883180541367896132143, −2.81740746489390912609286798375, −1.46805322910689172526787612745, −0.55888990023084356851247044929, 0.55888990023084356851247044929, 1.46805322910689172526787612745, 2.81740746489390912609286798375, 4.00130678883180541367896132143, 4.95274792428955110310719918564, 5.70777812953831334605653559952, 6.10856795548362819651765026537, 6.94953239261396067519141721208, 7.63138992022474016947705475199, 8.308213556642795604391394662977

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