Properties

Label 2-6018-1.1-c1-0-95
Degree $2$
Conductor $6018$
Sign $1$
Analytic cond. $48.0539$
Root an. cond. $6.93209$
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 + 3-s + 4-s + 3.56·5-s + 6-s − 2.46·7-s + 8-s + 9-s + 3.56·10-s + 2.23·11-s + 12-s + 6.98·13-s − 2.46·14-s + 3.56·15-s + 16-s − 17-s + 18-s − 3.26·19-s + 3.56·20-s − 2.46·21-s + 2.23·22-s − 0.970·23-s + 24-s + 7.70·25-s + 6.98·26-s + 27-s − 2.46·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.59·5-s + 0.408·6-s − 0.932·7-s + 0.353·8-s + 0.333·9-s + 1.12·10-s + 0.674·11-s + 0.288·12-s + 1.93·13-s − 0.659·14-s + 0.920·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.748·19-s + 0.796·20-s − 0.538·21-s + 0.476·22-s − 0.202·23-s + 0.204·24-s + 1.54·25-s + 1.37·26-s + 0.192·27-s − 0.466·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6018\)    =    \(2 \cdot 3 \cdot 17 \cdot 59\)
Sign: $1$
Analytic conductor: \(48.0539\)
Root analytic conductor: \(6.93209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6018,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.739343797\)
\(L(\frac12)\) \(\approx\) \(5.739343797\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
17 \( 1 + T \)
59 \( 1 - T \)
good5 \( 1 - 3.56T + 5T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 - 6.98T + 13T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 + 0.970T + 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
31 \( 1 - 2.91T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 + 5.50T + 41T^{2} \)
43 \( 1 + 6.05T + 43T^{2} \)
47 \( 1 - 0.503T + 47T^{2} \)
53 \( 1 - 11.1T + 53T^{2} \)
61 \( 1 - 6.17T + 61T^{2} \)
67 \( 1 - 2.12T + 67T^{2} \)
71 \( 1 + 3.41T + 71T^{2} \)
73 \( 1 + 1.74T + 73T^{2} \)
79 \( 1 + 0.425T + 79T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 - 7.98T + 97T^{2} \)
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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

−8.408554025873444965394421228322, −6.83948671926808120626269189170, −6.60691559140846477140769313806, −6.03420450408170628172889541298, −5.37197432011141599675584458514, −4.29281395093519668684264845353, −3.57914468251365826980290066488, −2.87951464201862583474007186721, −1.96659329965428230990828543494, −1.23125530890269007388235354459, 1.23125530890269007388235354459, 1.96659329965428230990828543494, 2.87951464201862583474007186721, 3.57914468251365826980290066488, 4.29281395093519668684264845353, 5.37197432011141599675584458514, 6.03420450408170628172889541298, 6.60691559140846477140769313806, 6.83948671926808120626269189170, 8.408554025873444965394421228322

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