Properties

Label 2-4022-1.1-c1-0-41
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 − 1.41·3-s + 4-s − 3.55·5-s + 1.41·6-s − 3.54·7-s − 8-s − 1.00·9-s + 3.55·10-s + 4.87·11-s − 1.41·12-s − 3.10·13-s + 3.54·14-s + 5.00·15-s + 16-s − 4.40·17-s + 1.00·18-s + 1.30·19-s − 3.55·20-s + 5.00·21-s − 4.87·22-s − 3.42·23-s + 1.41·24-s + 7.60·25-s + 3.10·26-s + 5.65·27-s − 3.54·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.814·3-s + 0.5·4-s − 1.58·5-s + 0.576·6-s − 1.34·7-s − 0.353·8-s − 0.336·9-s + 1.12·10-s + 1.46·11-s − 0.407·12-s − 0.861·13-s + 0.948·14-s + 1.29·15-s + 0.250·16-s − 1.06·17-s + 0.237·18-s + 0.299·19-s − 0.793·20-s + 1.09·21-s − 1.03·22-s − 0.713·23-s + 0.288·24-s + 1.52·25-s + 0.609·26-s + 1.08·27-s − 0.670·28-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 \( 1 + 1.41T + 3T^{2} \)
5 \( 1 + 3.55T + 5T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 - 4.87T + 11T^{2} \)
13 \( 1 + 3.10T + 13T^{2} \)
17 \( 1 + 4.40T + 17T^{2} \)
19 \( 1 - 1.30T + 19T^{2} \)
23 \( 1 + 3.42T + 23T^{2} \)
29 \( 1 - 8.36T + 29T^{2} \)
31 \( 1 + 3.07T + 31T^{2} \)
37 \( 1 - 7.52T + 37T^{2} \)
41 \( 1 + 4.62T + 41T^{2} \)
43 \( 1 + 4.44T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 8.60T + 53T^{2} \)
59 \( 1 - 14.4T + 59T^{2} \)
61 \( 1 - 7.09T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 + 6.84T + 83T^{2} \)
89 \( 1 + 2.84T + 89T^{2} \)
97 \( 1 - 0.0168T + 97T^{2} \)
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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

−8.333415807601977326155997183546, −7.08194767280800414547314238895, −6.77504333166065072206743368464, −6.23132882527228251618015607654, −5.08087201687663732776694025931, −4.12637485274196318309568574880, −3.48476089610824896840666954413, −2.51307324821246515238699623249, −0.792574291436769231333454711578, 0, 0.792574291436769231333454711578, 2.51307324821246515238699623249, 3.48476089610824896840666954413, 4.12637485274196318309568574880, 5.08087201687663732776694025931, 6.23132882527228251618015607654, 6.77504333166065072206743368464, 7.08194767280800414547314238895, 8.333415807601977326155997183546

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