Properties

Label 2-6018-1.1-c1-0-102
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 1.56·5-s + 6-s + 1.56·7-s − 8-s + 9-s + 1.56·10-s + 4.42·11-s − 12-s + 5.02·13-s − 1.56·14-s + 1.56·15-s + 16-s + 17-s − 18-s − 6.32·19-s − 1.56·20-s − 1.56·21-s − 4.42·22-s + 4.03·23-s + 24-s − 2.53·25-s − 5.02·26-s − 27-s + 1.56·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.702·5-s + 0.408·6-s + 0.592·7-s − 0.353·8-s + 0.333·9-s + 0.496·10-s + 1.33·11-s − 0.288·12-s + 1.39·13-s − 0.419·14-s + 0.405·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.45·19-s − 0.351·20-s − 0.342·21-s − 0.944·22-s + 0.841·23-s + 0.204·24-s − 0.507·25-s − 0.984·26-s − 0.192·27-s + 0.296·28-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: \(1\)
Selberg data: \((2,\ 6018,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 - 4.42T + 11T^{2} \)
13 \( 1 - 5.02T + 13T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 - 4.03T + 23T^{2} \)
29 \( 1 + 5.32T + 29T^{2} \)
31 \( 1 + 9.38T + 31T^{2} \)
37 \( 1 - 8.41T + 37T^{2} \)
41 \( 1 + 0.858T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + 2.96T + 47T^{2} \)
53 \( 1 - 0.373T + 53T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 + 7.36T + 67T^{2} \)
71 \( 1 + 1.31T + 71T^{2} \)
73 \( 1 - 0.0349T + 73T^{2} \)
79 \( 1 - 3.92T + 79T^{2} \)
83 \( 1 - 8.34T + 83T^{2} \)
89 \( 1 - 9.55T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−7.82632565028592292637238447169, −7.04229633368426702557506366835, −6.36876159401919766265349978296, −5.84408929956564643971976712306, −4.77446942548085288744309798025, −3.94922023753383541383695831750, −3.44090763669101222503380171253, −1.86563163062318753829452358823, −1.26715072178102289804394692514, 0, 1.26715072178102289804394692514, 1.86563163062318753829452358823, 3.44090763669101222503380171253, 3.94922023753383541383695831750, 4.77446942548085288744309798025, 5.84408929956564643971976712306, 6.36876159401919766265349978296, 7.04229633368426702557506366835, 7.82632565028592292637238447169

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