Properties

Label 2-4022-1.1-c1-0-33
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 + 0.590·3-s + 4-s + 3.96·5-s − 0.590·6-s − 4.53·7-s − 8-s − 2.65·9-s − 3.96·10-s − 5.22·11-s + 0.590·12-s − 0.575·13-s + 4.53·14-s + 2.34·15-s + 16-s − 2.92·17-s + 2.65·18-s + 5.48·19-s + 3.96·20-s − 2.67·21-s + 5.22·22-s + 4.54·23-s − 0.590·24-s + 10.7·25-s + 0.575·26-s − 3.33·27-s − 4.53·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.340·3-s + 0.5·4-s + 1.77·5-s − 0.241·6-s − 1.71·7-s − 0.353·8-s − 0.883·9-s − 1.25·10-s − 1.57·11-s + 0.170·12-s − 0.159·13-s + 1.21·14-s + 0.604·15-s + 0.250·16-s − 0.709·17-s + 0.624·18-s + 1.25·19-s + 0.886·20-s − 0.584·21-s + 1.11·22-s + 0.948·23-s − 0.120·24-s + 2.14·25-s + 0.112·26-s − 0.642·27-s − 0.856·28-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

\(L(1)\) \(\approx\) \(1.299424625\)
\(L(\frac12)\) \(\approx\) \(1.299424625\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.590T + 3T^{2} \)
5 \( 1 - 3.96T + 5T^{2} \)
7 \( 1 + 4.53T + 7T^{2} \)
11 \( 1 + 5.22T + 11T^{2} \)
13 \( 1 + 0.575T + 13T^{2} \)
17 \( 1 + 2.92T + 17T^{2} \)
19 \( 1 - 5.48T + 19T^{2} \)
23 \( 1 - 4.54T + 23T^{2} \)
29 \( 1 - 1.56T + 29T^{2} \)
31 \( 1 + 0.117T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 9.06T + 41T^{2} \)
43 \( 1 - 3.86T + 43T^{2} \)
47 \( 1 + 4.03T + 47T^{2} \)
53 \( 1 - 4.25T + 53T^{2} \)
59 \( 1 - 3.55T + 59T^{2} \)
61 \( 1 - 1.06T + 61T^{2} \)
67 \( 1 - 2.84T + 67T^{2} \)
71 \( 1 - 8.41T + 71T^{2} \)
73 \( 1 - 1.78T + 73T^{2} \)
79 \( 1 + 6.20T + 79T^{2} \)
83 \( 1 - 4.25T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 - 6.76T + 97T^{2} \)
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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

−8.701303505999355665683445559520, −7.78515846435636082936333377559, −6.89315768975240701623837506663, −6.29868437504582895565240508768, −5.62731720120797373432259840065, −5.07094567967195794364333502597, −3.26387440453098128760580690234, −2.74560556430492763947245099556, −2.22630032738932199447365912289, −0.67210535882495559166870142994, 0.67210535882495559166870142994, 2.22630032738932199447365912289, 2.74560556430492763947245099556, 3.26387440453098128760580690234, 5.07094567967195794364333502597, 5.62731720120797373432259840065, 6.29868437504582895565240508768, 6.89315768975240701623837506663, 7.78515846435636082936333377559, 8.701303505999355665683445559520

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