Properties

Label 2-6021-1.1-c1-0-94
Degree $2$
Conductor $6021$
Sign $1$
Analytic cond. $48.0779$
Root an. cond. $6.93382$
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  − 1.37·2-s − 0.103·4-s + 2.60·5-s − 2.14·7-s + 2.89·8-s − 3.58·10-s + 5.33·11-s − 0.800·13-s + 2.94·14-s − 3.78·16-s + 4.24·17-s − 3.39·19-s − 0.270·20-s − 7.34·22-s + 0.350·23-s + 1.78·25-s + 1.10·26-s + 0.222·28-s + 6.14·29-s + 1.59·31-s − 0.586·32-s − 5.85·34-s − 5.57·35-s + 1.51·37-s + 4.67·38-s + 7.54·40-s + 4.20·41-s + ⋯
L(s)  = 1  − 0.973·2-s − 0.0518·4-s + 1.16·5-s − 0.809·7-s + 1.02·8-s − 1.13·10-s + 1.60·11-s − 0.222·13-s + 0.787·14-s − 0.945·16-s + 1.03·17-s − 0.779·19-s − 0.0604·20-s − 1.56·22-s + 0.0730·23-s + 0.357·25-s + 0.216·26-s + 0.0419·28-s + 1.14·29-s + 0.286·31-s − 0.103·32-s − 1.00·34-s − 0.942·35-s + 0.248·37-s + 0.758·38-s + 1.19·40-s + 0.657·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6021\)    =    \(3^{3} \cdot 223\)
Sign: $1$
Analytic conductor: \(48.0779\)
Root analytic conductor: \(6.93382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6021,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418980170\)
\(L(\frac12)\) \(\approx\) \(1.418980170\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 - T \)
good2 \( 1 + 1.37T + 2T^{2} \)
5 \( 1 - 2.60T + 5T^{2} \)
7 \( 1 + 2.14T + 7T^{2} \)
11 \( 1 - 5.33T + 11T^{2} \)
13 \( 1 + 0.800T + 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 + 3.39T + 19T^{2} \)
23 \( 1 - 0.350T + 23T^{2} \)
29 \( 1 - 6.14T + 29T^{2} \)
31 \( 1 - 1.59T + 31T^{2} \)
37 \( 1 - 1.51T + 37T^{2} \)
41 \( 1 - 4.20T + 41T^{2} \)
43 \( 1 - 5.59T + 43T^{2} \)
47 \( 1 + 0.326T + 47T^{2} \)
53 \( 1 + 2.42T + 53T^{2} \)
59 \( 1 + 0.747T + 59T^{2} \)
61 \( 1 + 2.16T + 61T^{2} \)
67 \( 1 - 2.07T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 - 0.550T + 73T^{2} \)
79 \( 1 - 6.13T + 79T^{2} \)
83 \( 1 - 1.55T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 - 3.09T + 97T^{2} \)
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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

−8.253453948839963244441838655618, −7.46590874507355346590696238907, −6.48266485096123777571229081896, −6.32747028877466441970923087356, −5.31832217584432810759649876671, −4.40756712346386545199176910668, −3.62908958224649587069002697231, −2.52982504705277272407050966334, −1.56651616544300235483370160767, −0.78837303159778296676396997837, 0.78837303159778296676396997837, 1.56651616544300235483370160767, 2.52982504705277272407050966334, 3.62908958224649587069002697231, 4.40756712346386545199176910668, 5.31832217584432810759649876671, 6.32747028877466441970923087356, 6.48266485096123777571229081896, 7.46590874507355346590696238907, 8.253453948839963244441838655618

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