Properties

Label 2-6018-1.1-c1-0-81
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 1.47·5-s + 6-s − 0.770·7-s + 8-s + 9-s + 1.47·10-s + 2.85·11-s + 12-s + 4.25·13-s − 0.770·14-s + 1.47·15-s + 16-s + 17-s + 18-s + 0.374·19-s + 1.47·20-s − 0.770·21-s + 2.85·22-s + 0.425·23-s + 24-s − 2.82·25-s + 4.25·26-s + 27-s − 0.770·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.659·5-s + 0.408·6-s − 0.291·7-s + 0.353·8-s + 0.333·9-s + 0.466·10-s + 0.860·11-s + 0.288·12-s + 1.18·13-s − 0.205·14-s + 0.380·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.0858·19-s + 0.329·20-s − 0.168·21-s + 0.608·22-s + 0.0888·23-s + 0.204·24-s − 0.564·25-s + 0.834·26-s + 0.192·27-s − 0.145·28-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

\(L(1)\) \(\approx\) \(5.187622049\)
\(L(\frac12)\) \(\approx\) \(5.187622049\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.47T + 5T^{2} \)
7 \( 1 + 0.770T + 7T^{2} \)
11 \( 1 - 2.85T + 11T^{2} \)
13 \( 1 - 4.25T + 13T^{2} \)
19 \( 1 - 0.374T + 19T^{2} \)
23 \( 1 - 0.425T + 23T^{2} \)
29 \( 1 + 1.19T + 29T^{2} \)
31 \( 1 + 2.96T + 31T^{2} \)
37 \( 1 - 6.00T + 37T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 - 0.816T + 43T^{2} \)
47 \( 1 + 0.227T + 47T^{2} \)
53 \( 1 + 1.29T + 53T^{2} \)
61 \( 1 - 8.62T + 61T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 - 1.56T + 73T^{2} \)
79 \( 1 - 0.251T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 3.50T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.102570044493055011344918339001, −7.23242120417753076480276206952, −6.50246740127851645343726917733, −5.97968790719381891997063988711, −5.30120980124290771466811261960, −4.18805759403164163051916326091, −3.72960245716365426578509602460, −2.90387830257785157057806694601, −1.96852118218628641065307880619, −1.14814927712675766038910427096, 1.14814927712675766038910427096, 1.96852118218628641065307880619, 2.90387830257785157057806694601, 3.72960245716365426578509602460, 4.18805759403164163051916326091, 5.30120980124290771466811261960, 5.97968790719381891997063988711, 6.50246740127851645343726917733, 7.23242120417753076480276206952, 8.102570044493055011344918339001

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