Properties

Label 2-4022-1.1-c1-0-117
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 0.995·3-s + 4-s + 3.65·5-s + 0.995·6-s + 3.82·7-s + 8-s − 2.00·9-s + 3.65·10-s + 1.37·11-s + 0.995·12-s + 0.314·13-s + 3.82·14-s + 3.64·15-s + 16-s − 0.627·17-s − 2.00·18-s − 5.51·19-s + 3.65·20-s + 3.80·21-s + 1.37·22-s + 3.26·23-s + 0.995·24-s + 8.36·25-s + 0.314·26-s − 4.98·27-s + 3.82·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.574·3-s + 0.5·4-s + 1.63·5-s + 0.406·6-s + 1.44·7-s + 0.353·8-s − 0.669·9-s + 1.15·10-s + 0.415·11-s + 0.287·12-s + 0.0873·13-s + 1.02·14-s + 0.939·15-s + 0.250·16-s − 0.152·17-s − 0.473·18-s − 1.26·19-s + 0.817·20-s + 0.831·21-s + 0.294·22-s + 0.679·23-s + 0.203·24-s + 1.67·25-s + 0.0617·26-s − 0.959·27-s + 0.722·28-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

\(L(1)\) \(\approx\) \(5.789021945\)
\(L(\frac12)\) \(\approx\) \(5.789021945\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.995T + 3T^{2} \)
5 \( 1 - 3.65T + 5T^{2} \)
7 \( 1 - 3.82T + 7T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 - 0.314T + 13T^{2} \)
17 \( 1 + 0.627T + 17T^{2} \)
19 \( 1 + 5.51T + 19T^{2} \)
23 \( 1 - 3.26T + 23T^{2} \)
29 \( 1 + 4.51T + 29T^{2} \)
31 \( 1 + 0.743T + 31T^{2} \)
37 \( 1 - 3.92T + 37T^{2} \)
41 \( 1 + 4.88T + 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + 8.55T + 47T^{2} \)
53 \( 1 - 2.25T + 53T^{2} \)
59 \( 1 - 2.36T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 7.35T + 67T^{2} \)
71 \( 1 + 1.41T + 71T^{2} \)
73 \( 1 + 6.82T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 9.91T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 + 0.544T + 97T^{2} \)
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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

−8.583131992952776853222011186563, −7.73905144793910925439232573621, −6.81935029865729828514231208922, −6.01396100538937174664356269488, −5.49707854075591242862574164387, −4.77032754234713325252449761088, −3.92408723769212498757255173282, −2.72361317563570322563256122432, −2.11846512190846563985216954299, −1.42402589193504963188866504784, 1.42402589193504963188866504784, 2.11846512190846563985216954299, 2.72361317563570322563256122432, 3.92408723769212498757255173282, 4.77032754234713325252449761088, 5.49707854075591242862574164387, 6.01396100538937174664356269488, 6.81935029865729828514231208922, 7.73905144793910925439232573621, 8.583131992952776853222011186563

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