Properties

Label 2-6018-1.1-c1-0-33
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 − 0.621·5-s − 6-s + 2.55·7-s − 8-s + 9-s + 0.621·10-s + 0.810·11-s + 12-s − 0.218·13-s − 2.55·14-s − 0.621·15-s + 16-s + 17-s − 18-s − 7.58·19-s − 0.621·20-s + 2.55·21-s − 0.810·22-s − 3.76·23-s − 24-s − 4.61·25-s + 0.218·26-s + 27-s + 2.55·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.277·5-s − 0.408·6-s + 0.964·7-s − 0.353·8-s + 0.333·9-s + 0.196·10-s + 0.244·11-s + 0.288·12-s − 0.0605·13-s − 0.682·14-s − 0.160·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.73·19-s − 0.138·20-s + 0.557·21-s − 0.172·22-s − 0.784·23-s − 0.204·24-s − 0.922·25-s + 0.0428·26-s + 0.192·27-s + 0.482·28-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

\(L(1)\) \(\approx\) \(1.802715290\)
\(L(\frac12)\) \(\approx\) \(1.802715290\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.621T + 5T^{2} \)
7 \( 1 - 2.55T + 7T^{2} \)
11 \( 1 - 0.810T + 11T^{2} \)
13 \( 1 + 0.218T + 13T^{2} \)
19 \( 1 + 7.58T + 19T^{2} \)
23 \( 1 + 3.76T + 23T^{2} \)
29 \( 1 - 6.91T + 29T^{2} \)
31 \( 1 + 1.43T + 31T^{2} \)
37 \( 1 + 2.07T + 37T^{2} \)
41 \( 1 - 5.09T + 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 2.11T + 47T^{2} \)
53 \( 1 - 6.69T + 53T^{2} \)
61 \( 1 + 2.33T + 61T^{2} \)
67 \( 1 - 11.9T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 - 6.78T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 5.08T + 83T^{2} \)
89 \( 1 - 18.4T + 89T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−8.010613230104605754630165241904, −7.79945217505447874242679969413, −6.76761061732107275327045539035, −6.18526592500037716280226828841, −5.17140197269843749717428094103, −4.26195301396835247278301294550, −3.71245051086947769716863300357, −2.41593143244427406698702891629, −1.95275856808568802719210280750, −0.76340938214273792396636765566, 0.76340938214273792396636765566, 1.95275856808568802719210280750, 2.41593143244427406698702891629, 3.71245051086947769716863300357, 4.26195301396835247278301294550, 5.17140197269843749717428094103, 6.18526592500037716280226828841, 6.76761061732107275327045539035, 7.79945217505447874242679969413, 8.010613230104605754630165241904

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