Properties

Label 2-4022-1.1-c1-0-11
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 − 1.80·3-s + 4-s − 1.50·5-s + 1.80·6-s + 4.15·7-s − 8-s + 0.242·9-s + 1.50·10-s − 3.55·11-s − 1.80·12-s − 0.0848·13-s − 4.15·14-s + 2.70·15-s + 16-s − 6.67·17-s − 0.242·18-s − 7.45·19-s − 1.50·20-s − 7.47·21-s + 3.55·22-s − 1.44·23-s + 1.80·24-s − 2.73·25-s + 0.0848·26-s + 4.96·27-s + 4.15·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.03·3-s + 0.5·4-s − 0.672·5-s + 0.735·6-s + 1.56·7-s − 0.353·8-s + 0.0807·9-s + 0.475·10-s − 1.07·11-s − 0.519·12-s − 0.0235·13-s − 1.10·14-s + 0.699·15-s + 0.250·16-s − 1.61·17-s − 0.0570·18-s − 1.71·19-s − 0.336·20-s − 1.63·21-s + 0.758·22-s − 0.301·23-s + 0.367·24-s − 0.547·25-s + 0.0166·26-s + 0.955·27-s + 0.784·28-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

\(L(1)\) \(\approx\) \(0.3986204101\)
\(L(\frac12)\) \(\approx\) \(0.3986204101\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.80T + 3T^{2} \)
5 \( 1 + 1.50T + 5T^{2} \)
7 \( 1 - 4.15T + 7T^{2} \)
11 \( 1 + 3.55T + 11T^{2} \)
13 \( 1 + 0.0848T + 13T^{2} \)
17 \( 1 + 6.67T + 17T^{2} \)
19 \( 1 + 7.45T + 19T^{2} \)
23 \( 1 + 1.44T + 23T^{2} \)
29 \( 1 - 6.15T + 29T^{2} \)
31 \( 1 - 4.75T + 31T^{2} \)
37 \( 1 + 0.906T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 0.304T + 43T^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 - 7.20T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 5.00T + 61T^{2} \)
67 \( 1 + 2.29T + 67T^{2} \)
71 \( 1 + 5.84T + 71T^{2} \)
73 \( 1 + 8.74T + 73T^{2} \)
79 \( 1 - 5.64T + 79T^{2} \)
83 \( 1 - 2.64T + 83T^{2} \)
89 \( 1 - 4.52T + 89T^{2} \)
97 \( 1 - 5.91T + 97T^{2} \)
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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

−8.314120219218130711144448977470, −8.026355906482215821630875187561, −6.92160192674711643461050885665, −6.42063036107993781119098076842, −5.40865011616381884854796402809, −4.75776952860347336783340489388, −4.16708041057448955694690820603, −2.60335223963183517296716902185, −1.82218004892836926526381236234, −0.40948767978964373260282195901, 0.40948767978964373260282195901, 1.82218004892836926526381236234, 2.60335223963183517296716902185, 4.16708041057448955694690820603, 4.75776952860347336783340489388, 5.40865011616381884854796402809, 6.42063036107993781119098076842, 6.92160192674711643461050885665, 8.026355906482215821630875187561, 8.314120219218130711144448977470

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