Properties

Label 2-6021-1.1-c1-0-171
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  + 1.01·2-s − 0.976·4-s + 1.54·5-s + 4.83·7-s − 3.01·8-s + 1.55·10-s + 4.58·11-s + 0.937·13-s + 4.89·14-s − 1.09·16-s + 5.94·17-s + 1.55·19-s − 1.50·20-s + 4.63·22-s − 3.42·23-s − 2.62·25-s + 0.948·26-s − 4.71·28-s + 5.41·29-s − 0.550·31-s + 4.91·32-s + 6.01·34-s + 7.44·35-s + 3.01·37-s + 1.57·38-s − 4.63·40-s − 1.56·41-s + ⋯
L(s)  = 1  + 0.715·2-s − 0.488·4-s + 0.688·5-s + 1.82·7-s − 1.06·8-s + 0.492·10-s + 1.38·11-s + 0.260·13-s + 1.30·14-s − 0.273·16-s + 1.44·17-s + 0.357·19-s − 0.336·20-s + 0.989·22-s − 0.714·23-s − 0.525·25-s + 0.186·26-s − 0.891·28-s + 1.00·29-s − 0.0989·31-s + 0.868·32-s + 1.03·34-s + 1.25·35-s + 0.496·37-s + 0.256·38-s − 0.733·40-s − 0.244·41-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

\(L(1)\) \(\approx\) \(4.129111045\)
\(L(\frac12)\) \(\approx\) \(4.129111045\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.01T + 2T^{2} \)
5 \( 1 - 1.54T + 5T^{2} \)
7 \( 1 - 4.83T + 7T^{2} \)
11 \( 1 - 4.58T + 11T^{2} \)
13 \( 1 - 0.937T + 13T^{2} \)
17 \( 1 - 5.94T + 17T^{2} \)
19 \( 1 - 1.55T + 19T^{2} \)
23 \( 1 + 3.42T + 23T^{2} \)
29 \( 1 - 5.41T + 29T^{2} \)
31 \( 1 + 0.550T + 31T^{2} \)
37 \( 1 - 3.01T + 37T^{2} \)
41 \( 1 + 1.56T + 41T^{2} \)
43 \( 1 - 4.62T + 43T^{2} \)
47 \( 1 - 2.13T + 47T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + 6.96T + 59T^{2} \)
61 \( 1 + 6.57T + 61T^{2} \)
67 \( 1 - 6.39T + 67T^{2} \)
71 \( 1 + 4.75T + 71T^{2} \)
73 \( 1 + 9.95T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 9.08T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−8.077079195332003471822728401006, −7.49495655843511722712671891254, −6.29661313902853119663470068919, −5.83056696232945453207590696557, −5.15996690396837430363243942251, −4.45761959914270875730748133428, −3.90490015374090333973444919291, −2.93071229677090076167204800607, −1.70947372430643223141249581892, −1.10648665701176925270082229576, 1.10648665701176925270082229576, 1.70947372430643223141249581892, 2.93071229677090076167204800607, 3.90490015374090333973444919291, 4.45761959914270875730748133428, 5.15996690396837430363243942251, 5.83056696232945453207590696557, 6.29661313902853119663470068919, 7.49495655843511722712671891254, 8.077079195332003471822728401006

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