Properties

Label 2-6018-1.1-c1-0-46
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 − 0.760·5-s − 6-s − 1.62·7-s + 8-s + 9-s − 0.760·10-s + 3.63·11-s − 12-s + 6.76·13-s − 1.62·14-s + 0.760·15-s + 16-s − 17-s + 18-s + 2.10·19-s − 0.760·20-s + 1.62·21-s + 3.63·22-s + 7.89·23-s − 24-s − 4.42·25-s + 6.76·26-s − 27-s − 1.62·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.340·5-s − 0.408·6-s − 0.615·7-s + 0.353·8-s + 0.333·9-s − 0.240·10-s + 1.09·11-s − 0.288·12-s + 1.87·13-s − 0.434·14-s + 0.196·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.483·19-s − 0.170·20-s + 0.355·21-s + 0.773·22-s + 1.64·23-s − 0.204·24-s − 0.884·25-s + 1.32·26-s − 0.192·27-s − 0.307·28-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

\(L(1)\) \(\approx\) \(2.754213692\)
\(L(\frac12)\) \(\approx\) \(2.754213692\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.760T + 5T^{2} \)
7 \( 1 + 1.62T + 7T^{2} \)
11 \( 1 - 3.63T + 11T^{2} \)
13 \( 1 - 6.76T + 13T^{2} \)
19 \( 1 - 2.10T + 19T^{2} \)
23 \( 1 - 7.89T + 23T^{2} \)
29 \( 1 + 2.99T + 29T^{2} \)
31 \( 1 + 5.23T + 31T^{2} \)
37 \( 1 - 5.67T + 37T^{2} \)
41 \( 1 - 4.04T + 41T^{2} \)
43 \( 1 + 6.63T + 43T^{2} \)
47 \( 1 + 1.45T + 47T^{2} \)
53 \( 1 - 1.88T + 53T^{2} \)
61 \( 1 - 13.6T + 61T^{2} \)
67 \( 1 + 12.0T + 67T^{2} \)
71 \( 1 - 6.36T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 - 1.53T + 83T^{2} \)
89 \( 1 - 3.89T + 89T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−7.914537806997406680186173226199, −7.09644031687614684882746157736, −6.49661688364660256527186217009, −6.00149892136812385961283899071, −5.29573041597626413195969368930, −4.32145097915827910746162394166, −3.68717241947630413673568206723, −3.18795448617559548443531316696, −1.74940356495530072567829678744, −0.854593226449390796254641889613, 0.854593226449390796254641889613, 1.74940356495530072567829678744, 3.18795448617559548443531316696, 3.68717241947630413673568206723, 4.32145097915827910746162394166, 5.29573041597626413195969368930, 6.00149892136812385961283899071, 6.49661688364660256527186217009, 7.09644031687614684882746157736, 7.914537806997406680186173226199

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