Properties

Label 2-6018-1.1-c1-0-70
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 + 4.08·5-s + 6-s − 2.27·7-s + 8-s + 9-s + 4.08·10-s − 4.97·11-s + 12-s + 3.14·13-s − 2.27·14-s + 4.08·15-s + 16-s + 17-s + 18-s − 1.30·19-s + 4.08·20-s − 2.27·21-s − 4.97·22-s + 3.59·23-s + 24-s + 11.7·25-s + 3.14·26-s + 27-s − 2.27·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.82·5-s + 0.408·6-s − 0.860·7-s + 0.353·8-s + 0.333·9-s + 1.29·10-s − 1.50·11-s + 0.288·12-s + 0.872·13-s − 0.608·14-s + 1.05·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.299·19-s + 0.913·20-s − 0.497·21-s − 1.06·22-s + 0.749·23-s + 0.204·24-s + 2.34·25-s + 0.616·26-s + 0.192·27-s − 0.430·28-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

\(L(1)\) \(\approx\) \(5.290450319\)
\(L(\frac12)\) \(\approx\) \(5.290450319\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.08T + 5T^{2} \)
7 \( 1 + 2.27T + 7T^{2} \)
11 \( 1 + 4.97T + 11T^{2} \)
13 \( 1 - 3.14T + 13T^{2} \)
19 \( 1 + 1.30T + 19T^{2} \)
23 \( 1 - 3.59T + 23T^{2} \)
29 \( 1 - 0.323T + 29T^{2} \)
31 \( 1 + 1.53T + 31T^{2} \)
37 \( 1 - 6.77T + 37T^{2} \)
41 \( 1 - 8.35T + 41T^{2} \)
43 \( 1 - 5.11T + 43T^{2} \)
47 \( 1 - 3.42T + 47T^{2} \)
53 \( 1 - 2.94T + 53T^{2} \)
61 \( 1 + 7.45T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 6.20T + 71T^{2} \)
73 \( 1 - 8.75T + 73T^{2} \)
79 \( 1 - 9.24T + 79T^{2} \)
83 \( 1 - 1.66T + 83T^{2} \)
89 \( 1 + 3.42T + 89T^{2} \)
97 \( 1 + 8.73T + 97T^{2} \)
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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

−8.000492771401695323543560827988, −7.25848321940788643326042929764, −6.36374744717229756068014374097, −5.95970081059327542301383211793, −5.34159630639209008131887201299, −4.53627006012755948114519278678, −3.42355537105863708845211374580, −2.69407542865908233559213115753, −2.26363054015682631634507334793, −1.10534896426119940003556916700, 1.10534896426119940003556916700, 2.26363054015682631634507334793, 2.69407542865908233559213115753, 3.42355537105863708845211374580, 4.53627006012755948114519278678, 5.34159630639209008131887201299, 5.95970081059327542301383211793, 6.36374744717229756068014374097, 7.25848321940788643326042929764, 8.000492771401695323543560827988

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