Properties

Label 2-6020-1.1-c1-0-47
Degree $2$
Conductor $6020$
Sign $1$
Analytic cond. $48.0699$
Root an. cond. $6.93324$
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  + 3.12·3-s − 5-s − 7-s + 6.76·9-s + 5.76·11-s + 1.18·13-s − 3.12·15-s − 7.93·17-s − 0.642·19-s − 3.12·21-s + 7.81·23-s + 25-s + 11.7·27-s + 9.81·29-s − 8.66·31-s + 18.0·33-s + 35-s − 6.80·37-s + 3.71·39-s + 7.47·41-s − 43-s − 6.76·45-s + 12.6·47-s + 49-s − 24.7·51-s − 8.21·53-s − 5.76·55-s + ⋯
L(s)  = 1  + 1.80·3-s − 0.447·5-s − 0.377·7-s + 2.25·9-s + 1.73·11-s + 0.329·13-s − 0.806·15-s − 1.92·17-s − 0.147·19-s − 0.681·21-s + 1.62·23-s + 0.200·25-s + 2.26·27-s + 1.82·29-s − 1.55·31-s + 3.13·33-s + 0.169·35-s − 1.11·37-s + 0.594·39-s + 1.16·41-s − 0.152·43-s − 1.00·45-s + 1.84·47-s + 0.142·49-s − 3.47·51-s − 1.12·53-s − 0.777·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6020\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 43\)
Sign: $1$
Analytic conductor: \(48.0699\)
Root analytic conductor: \(6.93324\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6020,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.139103577\)
\(L(\frac12)\) \(\approx\) \(4.139103577\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
43 \( 1 + T \)
good3 \( 1 - 3.12T + 3T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
13 \( 1 - 1.18T + 13T^{2} \)
17 \( 1 + 7.93T + 17T^{2} \)
19 \( 1 + 0.642T + 19T^{2} \)
23 \( 1 - 7.81T + 23T^{2} \)
29 \( 1 - 9.81T + 29T^{2} \)
31 \( 1 + 8.66T + 31T^{2} \)
37 \( 1 + 6.80T + 37T^{2} \)
41 \( 1 - 7.47T + 41T^{2} \)
47 \( 1 - 12.6T + 47T^{2} \)
53 \( 1 + 8.21T + 53T^{2} \)
59 \( 1 - 0.705T + 59T^{2} \)
61 \( 1 + 4.03T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 - 4.05T + 71T^{2} \)
73 \( 1 - 4.31T + 73T^{2} \)
79 \( 1 - 5.17T + 79T^{2} \)
83 \( 1 - 6.02T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + 5.87T + 97T^{2} \)
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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

−8.361995713259788795236297939836, −7.28944822052882899370370256869, −6.89822682132430768840314488221, −6.32932756688397254105486155082, −4.83278749025996076884557400664, −4.11297248916598961550697372561, −3.65143756321169627397142091068, −2.85278034584286698614528509077, −2.03686895084713465009380403961, −1.03878190212523688362476362706, 1.03878190212523688362476362706, 2.03686895084713465009380403961, 2.85278034584286698614528509077, 3.65143756321169627397142091068, 4.11297248916598961550697372561, 4.83278749025996076884557400664, 6.32932756688397254105486155082, 6.89822682132430768840314488221, 7.28944822052882899370370256869, 8.361995713259788795236297939836

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