Properties

Label 2-4022-1.1-c1-0-1
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 − 0.923·3-s + 4-s − 1.00·5-s + 0.923·6-s − 1.65·7-s − 8-s − 2.14·9-s + 1.00·10-s − 3.52·11-s − 0.923·12-s − 2.87·13-s + 1.65·14-s + 0.928·15-s + 16-s − 0.517·17-s + 2.14·18-s − 2.41·19-s − 1.00·20-s + 1.52·21-s + 3.52·22-s − 7.18·23-s + 0.923·24-s − 3.98·25-s + 2.87·26-s + 4.75·27-s − 1.65·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.533·3-s + 0.5·4-s − 0.449·5-s + 0.377·6-s − 0.625·7-s − 0.353·8-s − 0.715·9-s + 0.317·10-s − 1.06·11-s − 0.266·12-s − 0.796·13-s + 0.442·14-s + 0.239·15-s + 0.250·16-s − 0.125·17-s + 0.505·18-s − 0.554·19-s − 0.224·20-s + 0.333·21-s + 0.752·22-s − 1.49·23-s + 0.188·24-s − 0.797·25-s + 0.563·26-s + 0.915·27-s − 0.312·28-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

\(L(1)\) \(\approx\) \(0.02115975205\)
\(L(\frac12)\) \(\approx\) \(0.02115975205\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.923T + 3T^{2} \)
5 \( 1 + 1.00T + 5T^{2} \)
7 \( 1 + 1.65T + 7T^{2} \)
11 \( 1 + 3.52T + 11T^{2} \)
13 \( 1 + 2.87T + 13T^{2} \)
17 \( 1 + 0.517T + 17T^{2} \)
19 \( 1 + 2.41T + 19T^{2} \)
23 \( 1 + 7.18T + 23T^{2} \)
29 \( 1 - 2.16T + 29T^{2} \)
31 \( 1 + 6.60T + 31T^{2} \)
37 \( 1 + 9.53T + 37T^{2} \)
41 \( 1 + 4.03T + 41T^{2} \)
43 \( 1 - 9.99T + 43T^{2} \)
47 \( 1 - 1.45T + 47T^{2} \)
53 \( 1 + 7.71T + 53T^{2} \)
59 \( 1 + 6.62T + 59T^{2} \)
61 \( 1 - 5.00T + 61T^{2} \)
67 \( 1 + 8.51T + 67T^{2} \)
71 \( 1 - 3.57T + 71T^{2} \)
73 \( 1 - 0.496T + 73T^{2} \)
79 \( 1 + 3.81T + 79T^{2} \)
83 \( 1 + 4.39T + 83T^{2} \)
89 \( 1 + 7.28T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.374434385205459911742885591905, −7.77388175757366504463031644218, −7.11976254085594998781472875868, −6.21808220236356313278040871216, −5.66451862127044204105371587590, −4.81597278308507150860642670652, −3.72472868104478188195482036938, −2.80748427126123209962720865659, −1.95396395408978061087424711414, −0.087578655656298897786562394226, 0.087578655656298897786562394226, 1.95396395408978061087424711414, 2.80748427126123209962720865659, 3.72472868104478188195482036938, 4.81597278308507150860642670652, 5.66451862127044204105371587590, 6.21808220236356313278040871216, 7.11976254085594998781472875868, 7.77388175757366504463031644218, 8.374434385205459911742885591905

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