Properties

Label 2-4022-1.1-c1-0-50
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 − 0.0611·3-s + 4-s − 3.49·5-s + 0.0611·6-s − 5.08·7-s − 8-s − 2.99·9-s + 3.49·10-s − 0.435·11-s − 0.0611·12-s + 0.685·13-s + 5.08·14-s + 0.214·15-s + 16-s + 4.65·17-s + 2.99·18-s + 8.26·19-s − 3.49·20-s + 0.310·21-s + 0.435·22-s − 1.27·23-s + 0.0611·24-s + 7.24·25-s − 0.685·26-s + 0.366·27-s − 5.08·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0353·3-s + 0.5·4-s − 1.56·5-s + 0.0249·6-s − 1.92·7-s − 0.353·8-s − 0.998·9-s + 1.10·10-s − 0.131·11-s − 0.0176·12-s + 0.190·13-s + 1.35·14-s + 0.0552·15-s + 0.250·16-s + 1.13·17-s + 0.706·18-s + 1.89·19-s − 0.782·20-s + 0.0678·21-s + 0.0929·22-s − 0.266·23-s + 0.0124·24-s + 1.44·25-s − 0.134·26-s + 0.0706·27-s − 0.960·28-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 \( 1 + 0.0611T + 3T^{2} \)
5 \( 1 + 3.49T + 5T^{2} \)
7 \( 1 + 5.08T + 7T^{2} \)
11 \( 1 + 0.435T + 11T^{2} \)
13 \( 1 - 0.685T + 13T^{2} \)
17 \( 1 - 4.65T + 17T^{2} \)
19 \( 1 - 8.26T + 19T^{2} \)
23 \( 1 + 1.27T + 23T^{2} \)
29 \( 1 + 0.893T + 29T^{2} \)
31 \( 1 - 4.10T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 - 8.20T + 41T^{2} \)
43 \( 1 + 3.04T + 43T^{2} \)
47 \( 1 - 4.10T + 47T^{2} \)
53 \( 1 - 9.92T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 8.02T + 67T^{2} \)
71 \( 1 - 1.52T + 71T^{2} \)
73 \( 1 - 3.89T + 73T^{2} \)
79 \( 1 - 0.879T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 + 2.11T + 89T^{2} \)
97 \( 1 - 1.77T + 97T^{2} \)
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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

−7.905679887838051332954551462166, −7.56843770944773302605886582953, −6.79915895717850537618644627271, −5.99106442641228098720847911552, −5.27927733769458058940339933850, −3.86183372897678411357289046055, −3.25260938467129038128118353125, −2.86044524620644672037014714635, −0.883195815706901430124528971916, 0, 0.883195815706901430124528971916, 2.86044524620644672037014714635, 3.25260938467129038128118353125, 3.86183372897678411357289046055, 5.27927733769458058940339933850, 5.99106442641228098720847911552, 6.79915895717850537618644627271, 7.56843770944773302605886582953, 7.905679887838051332954551462166

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