Properties

Label 2-4022-1.1-c1-0-71
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.37·3-s + 4-s − 1.97·5-s − 2.37·6-s − 4.20·7-s + 8-s + 2.64·9-s − 1.97·10-s − 5.02·11-s − 2.37·12-s + 5.57·13-s − 4.20·14-s + 4.69·15-s + 16-s + 6.46·17-s + 2.64·18-s + 4.71·19-s − 1.97·20-s + 9.99·21-s − 5.02·22-s + 1.78·23-s − 2.37·24-s − 1.09·25-s + 5.57·26-s + 0.850·27-s − 4.20·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.37·3-s + 0.5·4-s − 0.884·5-s − 0.969·6-s − 1.58·7-s + 0.353·8-s + 0.880·9-s − 0.625·10-s − 1.51·11-s − 0.685·12-s + 1.54·13-s − 1.12·14-s + 1.21·15-s + 0.250·16-s + 1.56·17-s + 0.622·18-s + 1.08·19-s − 0.442·20-s + 2.18·21-s − 1.07·22-s + 0.372·23-s − 0.484·24-s − 0.218·25-s + 1.09·26-s + 0.163·27-s − 0.794·28-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: \(1\)
Selberg data: \((2,\ 4022,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 3T^{2} \)
5 \( 1 + 1.97T + 5T^{2} \)
7 \( 1 + 4.20T + 7T^{2} \)
11 \( 1 + 5.02T + 11T^{2} \)
13 \( 1 - 5.57T + 13T^{2} \)
17 \( 1 - 6.46T + 17T^{2} \)
19 \( 1 - 4.71T + 19T^{2} \)
23 \( 1 - 1.78T + 23T^{2} \)
29 \( 1 + 7.59T + 29T^{2} \)
31 \( 1 - 3.98T + 31T^{2} \)
37 \( 1 - 5.87T + 37T^{2} \)
41 \( 1 + 1.51T + 41T^{2} \)
43 \( 1 - 3.13T + 43T^{2} \)
47 \( 1 - 0.621T + 47T^{2} \)
53 \( 1 - 0.0835T + 53T^{2} \)
59 \( 1 - 0.667T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 + 12.5T + 67T^{2} \)
71 \( 1 + 2.09T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 - 8.22T + 89T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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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

−7.55540065751776786232223092324, −7.43469587560505538296428775271, −6.14169632393828401063963243127, −5.93375470743037077738099462851, −5.30399833829285859479291254452, −4.30960922780176772525971007064, −3.36454040157008236017297734388, −3.01139098954918188212704874957, −1.07565008409424349464101738992, 0, 1.07565008409424349464101738992, 3.01139098954918188212704874957, 3.36454040157008236017297734388, 4.30960922780176772525971007064, 5.30399833829285859479291254452, 5.93375470743037077738099462851, 6.14169632393828401063963243127, 7.43469587560505538296428775271, 7.55540065751776786232223092324

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