Properties

Label 2-4022-1.1-c1-0-48
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 − 0.961·3-s + 4-s + 0.960·5-s + 0.961·6-s − 1.91·7-s − 8-s − 2.07·9-s − 0.960·10-s + 4.70·11-s − 0.961·12-s + 6.66·13-s + 1.91·14-s − 0.923·15-s + 16-s + 2.88·17-s + 2.07·18-s + 7.64·19-s + 0.960·20-s + 1.84·21-s − 4.70·22-s + 2.63·23-s + 0.961·24-s − 4.07·25-s − 6.66·26-s + 4.88·27-s − 1.91·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.555·3-s + 0.5·4-s + 0.429·5-s + 0.392·6-s − 0.724·7-s − 0.353·8-s − 0.691·9-s − 0.303·10-s + 1.41·11-s − 0.277·12-s + 1.84·13-s + 0.512·14-s − 0.238·15-s + 0.250·16-s + 0.699·17-s + 0.489·18-s + 1.75·19-s + 0.214·20-s + 0.402·21-s − 1.00·22-s + 0.549·23-s + 0.196·24-s − 0.815·25-s − 1.30·26-s + 0.939·27-s − 0.362·28-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

\(L(1)\) \(\approx\) \(1.381280289\)
\(L(\frac12)\) \(\approx\) \(1.381280289\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.961T + 3T^{2} \)
5 \( 1 - 0.960T + 5T^{2} \)
7 \( 1 + 1.91T + 7T^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 - 6.66T + 13T^{2} \)
17 \( 1 - 2.88T + 17T^{2} \)
19 \( 1 - 7.64T + 19T^{2} \)
23 \( 1 - 2.63T + 23T^{2} \)
29 \( 1 - 8.27T + 29T^{2} \)
31 \( 1 - 1.02T + 31T^{2} \)
37 \( 1 - 4.09T + 37T^{2} \)
41 \( 1 + 7.88T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 - 2.44T + 47T^{2} \)
53 \( 1 + 5.49T + 53T^{2} \)
59 \( 1 + 1.03T + 59T^{2} \)
61 \( 1 + 6.38T + 61T^{2} \)
67 \( 1 + 3.67T + 67T^{2} \)
71 \( 1 + 16.3T + 71T^{2} \)
73 \( 1 - 9.81T + 73T^{2} \)
79 \( 1 - 5.53T + 79T^{2} \)
83 \( 1 + 1.75T + 83T^{2} \)
89 \( 1 - 1.03T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.639923694881220470622483499315, −7.81039918987625336128079563331, −6.76976368991838324808574869656, −6.22428630291425260803373193121, −5.88085128883042061865965767400, −4.84698241109693083129303692654, −3.42970616308233952144191654658, −3.18127966216338809303467553044, −1.52130511914594408359980107673, −0.856912932396458801940517506793, 0.856912932396458801940517506793, 1.52130511914594408359980107673, 3.18127966216338809303467553044, 3.42970616308233952144191654658, 4.84698241109693083129303692654, 5.88085128883042061865965767400, 6.22428630291425260803373193121, 6.76976368991838324808574869656, 7.81039918987625336128079563331, 8.639923694881220470622483499315

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