Properties

Label 2-6018-1.1-c1-0-72
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 + 1.44·5-s − 6-s + 4.37·7-s − 8-s + 9-s − 1.44·10-s + 5.18·11-s + 12-s − 1.77·13-s − 4.37·14-s + 1.44·15-s + 16-s + 17-s − 18-s + 1.63·19-s + 1.44·20-s + 4.37·21-s − 5.18·22-s + 0.245·23-s − 24-s − 2.89·25-s + 1.77·26-s + 27-s + 4.37·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.648·5-s − 0.408·6-s + 1.65·7-s − 0.353·8-s + 0.333·9-s − 0.458·10-s + 1.56·11-s + 0.288·12-s − 0.491·13-s − 1.17·14-s + 0.374·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.375·19-s + 0.324·20-s + 0.955·21-s − 1.10·22-s + 0.0511·23-s − 0.204·24-s − 0.579·25-s + 0.347·26-s + 0.192·27-s + 0.827·28-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

\(L(1)\) \(\approx\) \(2.989649185\)
\(L(\frac12)\) \(\approx\) \(2.989649185\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.44T + 5T^{2} \)
7 \( 1 - 4.37T + 7T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
13 \( 1 + 1.77T + 13T^{2} \)
19 \( 1 - 1.63T + 19T^{2} \)
23 \( 1 - 0.245T + 23T^{2} \)
29 \( 1 - 4.12T + 29T^{2} \)
31 \( 1 + 4.83T + 31T^{2} \)
37 \( 1 - 7.94T + 37T^{2} \)
41 \( 1 + 9.26T + 41T^{2} \)
43 \( 1 + 1.10T + 43T^{2} \)
47 \( 1 - 4.83T + 47T^{2} \)
53 \( 1 + 6.02T + 53T^{2} \)
61 \( 1 - 12.3T + 61T^{2} \)
67 \( 1 - 1.78T + 67T^{2} \)
71 \( 1 - 1.68T + 71T^{2} \)
73 \( 1 + 3.63T + 73T^{2} \)
79 \( 1 + 8.59T + 79T^{2} \)
83 \( 1 - 1.43T + 83T^{2} \)
89 \( 1 - 3.26T + 89T^{2} \)
97 \( 1 + 4.03T + 97T^{2} \)
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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

−8.224414117206517842765728323610, −7.49771699038887869085008422597, −6.89668368046569965660662254759, −6.04108228249851199694022752767, −5.22256068633124294120399310112, −4.42619889387049212823133698238, −3.59542751502539054921167745093, −2.43184786807255772901072729275, −1.71449420693554404211869624461, −1.11184534513317322919471253254, 1.11184534513317322919471253254, 1.71449420693554404211869624461, 2.43184786807255772901072729275, 3.59542751502539054921167745093, 4.42619889387049212823133698238, 5.22256068633124294120399310112, 6.04108228249851199694022752767, 6.89668368046569965660662254759, 7.49771699038887869085008422597, 8.224414117206517842765728323610

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