Properties

Label 2-4022-1.1-c1-0-138
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 − 1.16·3-s + 4-s + 3.05·5-s + 1.16·6-s + 2.13·7-s − 8-s − 1.64·9-s − 3.05·10-s + 2.96·11-s − 1.16·12-s − 2.55·13-s − 2.13·14-s − 3.55·15-s + 16-s + 0.966·17-s + 1.64·18-s − 1.03·19-s + 3.05·20-s − 2.48·21-s − 2.96·22-s − 7.18·23-s + 1.16·24-s + 4.35·25-s + 2.55·26-s + 5.40·27-s + 2.13·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.671·3-s + 0.5·4-s + 1.36·5-s + 0.474·6-s + 0.807·7-s − 0.353·8-s − 0.549·9-s − 0.966·10-s + 0.893·11-s − 0.335·12-s − 0.708·13-s − 0.571·14-s − 0.918·15-s + 0.250·16-s + 0.234·17-s + 0.388·18-s − 0.236·19-s + 0.683·20-s − 0.542·21-s − 0.632·22-s − 1.49·23-s + 0.237·24-s + 0.870·25-s + 0.501·26-s + 1.04·27-s + 0.403·28-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 \( 1 + 1.16T + 3T^{2} \)
5 \( 1 - 3.05T + 5T^{2} \)
7 \( 1 - 2.13T + 7T^{2} \)
11 \( 1 - 2.96T + 11T^{2} \)
13 \( 1 + 2.55T + 13T^{2} \)
17 \( 1 - 0.966T + 17T^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
23 \( 1 + 7.18T + 23T^{2} \)
29 \( 1 + 3.72T + 29T^{2} \)
31 \( 1 + 5.13T + 31T^{2} \)
37 \( 1 - 3.04T + 37T^{2} \)
41 \( 1 + 9.21T + 41T^{2} \)
43 \( 1 + 8.73T + 43T^{2} \)
47 \( 1 + 1.66T + 47T^{2} \)
53 \( 1 - 1.21T + 53T^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 4.34T + 61T^{2} \)
67 \( 1 - 6.10T + 67T^{2} \)
71 \( 1 + 1.70T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 + 5.88T + 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 + 0.694T + 89T^{2} \)
97 \( 1 - 3.16T + 97T^{2} \)
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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

−8.224823017782181898111830804759, −7.32831152772085139517840989867, −6.49100198083967446439469581385, −5.91387335733246083296374485859, −5.35911390527497015112284712825, −4.49294814359875909736209279943, −3.19274079067380770995775457032, −1.99287556329014184697603943368, −1.56054190415359343010079737959, 0, 1.56054190415359343010079737959, 1.99287556329014184697603943368, 3.19274079067380770995775457032, 4.49294814359875909736209279943, 5.35911390527497015112284712825, 5.91387335733246083296374485859, 6.49100198083967446439469581385, 7.32831152772085139517840989867, 8.224823017782181898111830804759

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