Properties

Label 2-4022-1.1-c1-0-96
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 + 2.44·3-s + 4-s + 3.50·5-s − 2.44·6-s + 1.21·7-s − 8-s + 2.96·9-s − 3.50·10-s + 1.41·11-s + 2.44·12-s + 4.25·13-s − 1.21·14-s + 8.54·15-s + 16-s − 6.92·17-s − 2.96·18-s + 1.70·19-s + 3.50·20-s + 2.95·21-s − 1.41·22-s − 6.96·23-s − 2.44·24-s + 7.25·25-s − 4.25·26-s − 0.0935·27-s + 1.21·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.40·3-s + 0.5·4-s + 1.56·5-s − 0.996·6-s + 0.457·7-s − 0.353·8-s + 0.987·9-s − 1.10·10-s + 0.426·11-s + 0.704·12-s + 1.18·13-s − 0.323·14-s + 2.20·15-s + 0.250·16-s − 1.67·17-s − 0.698·18-s + 0.392·19-s + 0.782·20-s + 0.645·21-s − 0.301·22-s − 1.45·23-s − 0.498·24-s + 1.45·25-s − 0.835·26-s − 0.0180·27-s + 0.228·28-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

\(L(1)\) \(\approx\) \(3.510281279\)
\(L(\frac12)\) \(\approx\) \(3.510281279\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.44T + 3T^{2} \)
5 \( 1 - 3.50T + 5T^{2} \)
7 \( 1 - 1.21T + 7T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 4.25T + 13T^{2} \)
17 \( 1 + 6.92T + 17T^{2} \)
19 \( 1 - 1.70T + 19T^{2} \)
23 \( 1 + 6.96T + 23T^{2} \)
29 \( 1 + 4.05T + 29T^{2} \)
31 \( 1 - 6.03T + 31T^{2} \)
37 \( 1 - 7.22T + 37T^{2} \)
41 \( 1 + 0.637T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 0.459T + 47T^{2} \)
53 \( 1 + 2.84T + 53T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 14.7T + 61T^{2} \)
67 \( 1 - 3.19T + 67T^{2} \)
71 \( 1 + 6.99T + 71T^{2} \)
73 \( 1 + 5.46T + 73T^{2} \)
79 \( 1 + 4.45T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 3.97T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−8.713204008511379801957441249649, −8.006005532436952417308195016654, −7.13615513947861326067997019457, −6.22003558029984015173015280447, −5.84913351341481041535616711418, −4.49925002876549548992951106861, −3.65555170723225414855085862378, −2.47228527864446783417911513125, −2.10800704575933077548817310567, −1.23357066029666010461398375250, 1.23357066029666010461398375250, 2.10800704575933077548817310567, 2.47228527864446783417911513125, 3.65555170723225414855085862378, 4.49925002876549548992951106861, 5.84913351341481041535616711418, 6.22003558029984015173015280447, 7.13615513947861326067997019457, 8.006005532436952417308195016654, 8.713204008511379801957441249649

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