Properties

Label 2-6021-1.1-c1-0-118
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  − 0.323·2-s − 1.89·4-s + 3.36·5-s + 3.36·7-s + 1.25·8-s − 1.08·10-s − 1.65·11-s + 4.48·13-s − 1.08·14-s + 3.38·16-s + 0.370·17-s − 3.44·19-s − 6.37·20-s + 0.536·22-s − 3.00·23-s + 6.31·25-s − 1.45·26-s − 6.37·28-s − 0.548·29-s + 2.39·31-s − 3.61·32-s − 0.119·34-s + 11.3·35-s − 2.38·37-s + 1.11·38-s + 4.23·40-s + 4.88·41-s + ⋯
L(s)  = 1  − 0.228·2-s − 0.947·4-s + 1.50·5-s + 1.27·7-s + 0.445·8-s − 0.344·10-s − 0.500·11-s + 1.24·13-s − 0.290·14-s + 0.845·16-s + 0.0897·17-s − 0.790·19-s − 1.42·20-s + 0.114·22-s − 0.625·23-s + 1.26·25-s − 0.284·26-s − 1.20·28-s − 0.101·29-s + 0.430·31-s − 0.638·32-s − 0.0205·34-s + 1.91·35-s − 0.391·37-s + 0.180·38-s + 0.670·40-s + 0.762·41-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

\(L(1)\) \(\approx\) \(2.425027154\)
\(L(\frac12)\) \(\approx\) \(2.425027154\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.323T + 2T^{2} \)
5 \( 1 - 3.36T + 5T^{2} \)
7 \( 1 - 3.36T + 7T^{2} \)
11 \( 1 + 1.65T + 11T^{2} \)
13 \( 1 - 4.48T + 13T^{2} \)
17 \( 1 - 0.370T + 17T^{2} \)
19 \( 1 + 3.44T + 19T^{2} \)
23 \( 1 + 3.00T + 23T^{2} \)
29 \( 1 + 0.548T + 29T^{2} \)
31 \( 1 - 2.39T + 31T^{2} \)
37 \( 1 + 2.38T + 37T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 - 8.31T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 7.04T + 53T^{2} \)
59 \( 1 + 2.72T + 59T^{2} \)
61 \( 1 + 0.972T + 61T^{2} \)
67 \( 1 + 7.10T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 5.65T + 73T^{2} \)
79 \( 1 - 7.20T + 79T^{2} \)
83 \( 1 - 5.70T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 3.30T + 97T^{2} \)
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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

−8.221489869001578358576453247663, −7.63348296483674861548500224744, −6.48943974800293855226493947039, −5.74476492900547490127636024148, −5.35792099750107917885103468534, −4.50246320703259346044904581927, −3.85331386529853152659511082259, −2.51067316790513673353993642412, −1.73452054768724283728299566653, −0.923432101294479770349458908882, 0.923432101294479770349458908882, 1.73452054768724283728299566653, 2.51067316790513673353993642412, 3.85331386529853152659511082259, 4.50246320703259346044904581927, 5.35792099750107917885103468534, 5.74476492900547490127636024148, 6.48943974800293855226493947039, 7.63348296483674861548500224744, 8.221489869001578358576453247663

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