Properties

Label 2-4022-1.1-c1-0-137
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 + 2.90·3-s + 4-s + 2.46·5-s + 2.90·6-s + 0.871·7-s + 8-s + 5.45·9-s + 2.46·10-s + 4.42·11-s + 2.90·12-s − 6.00·13-s + 0.871·14-s + 7.18·15-s + 16-s − 0.755·17-s + 5.45·18-s − 2.18·19-s + 2.46·20-s + 2.53·21-s + 4.42·22-s − 3.76·23-s + 2.90·24-s + 1.09·25-s − 6.00·26-s + 7.14·27-s + 0.871·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.67·3-s + 0.5·4-s + 1.10·5-s + 1.18·6-s + 0.329·7-s + 0.353·8-s + 1.81·9-s + 0.780·10-s + 1.33·11-s + 0.839·12-s − 1.66·13-s + 0.232·14-s + 1.85·15-s + 0.250·16-s − 0.183·17-s + 1.28·18-s − 0.500·19-s + 0.552·20-s + 0.553·21-s + 0.942·22-s − 0.785·23-s + 0.593·24-s + 0.219·25-s − 1.17·26-s + 1.37·27-s + 0.164·28-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

\(L(1)\) \(\approx\) \(7.112045822\)
\(L(\frac12)\) \(\approx\) \(7.112045822\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.90T + 3T^{2} \)
5 \( 1 - 2.46T + 5T^{2} \)
7 \( 1 - 0.871T + 7T^{2} \)
11 \( 1 - 4.42T + 11T^{2} \)
13 \( 1 + 6.00T + 13T^{2} \)
17 \( 1 + 0.755T + 17T^{2} \)
19 \( 1 + 2.18T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 - 2.75T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 3.33T + 43T^{2} \)
47 \( 1 - 8.80T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 - 9.07T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 - 4.06T + 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 9.72T + 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 1.35T + 83T^{2} \)
89 \( 1 + 9.23T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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

−8.571735440393639971708207082040, −7.52951649384554414040248578284, −7.18640740921897122008712714888, −6.19134724196344413822138279290, −5.46248777208273475769242716986, −4.29902177153614209988362575941, −3.98432300641992594411186235645, −2.73899289700243599231392219746, −2.24200575931739849892458048528, −1.55423189609727758173073634713, 1.55423189609727758173073634713, 2.24200575931739849892458048528, 2.73899289700243599231392219746, 3.98432300641992594411186235645, 4.29902177153614209988362575941, 5.46248777208273475769242716986, 6.19134724196344413822138279290, 7.18640740921897122008712714888, 7.52951649384554414040248578284, 8.571735440393639971708207082040

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