Properties

Label 2-6018-1.1-c1-0-91
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 − 0.266·5-s + 6-s + 0.661·7-s + 8-s + 9-s − 0.266·10-s + 6.00·11-s + 12-s + 4.92·13-s + 0.661·14-s − 0.266·15-s + 16-s − 17-s + 18-s + 2.93·19-s − 0.266·20-s + 0.661·21-s + 6.00·22-s + 3.23·23-s + 24-s − 4.92·25-s + 4.92·26-s + 27-s + 0.661·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.119·5-s + 0.408·6-s + 0.249·7-s + 0.353·8-s + 0.333·9-s − 0.0843·10-s + 1.81·11-s + 0.288·12-s + 1.36·13-s + 0.176·14-s − 0.0689·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s + 0.673·19-s − 0.0596·20-s + 0.144·21-s + 1.28·22-s + 0.675·23-s + 0.204·24-s − 0.985·25-s + 0.965·26-s + 0.192·27-s + 0.124·28-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

\(L(1)\) \(\approx\) \(5.145500060\)
\(L(\frac12)\) \(\approx\) \(5.145500060\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.266T + 5T^{2} \)
7 \( 1 - 0.661T + 7T^{2} \)
11 \( 1 - 6.00T + 11T^{2} \)
13 \( 1 - 4.92T + 13T^{2} \)
19 \( 1 - 2.93T + 19T^{2} \)
23 \( 1 - 3.23T + 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 3.09T + 31T^{2} \)
37 \( 1 + 8.66T + 37T^{2} \)
41 \( 1 + 2.52T + 41T^{2} \)
43 \( 1 - 7.24T + 43T^{2} \)
47 \( 1 + 7.06T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
61 \( 1 + 5.51T + 61T^{2} \)
67 \( 1 - 9.08T + 67T^{2} \)
71 \( 1 - 0.403T + 71T^{2} \)
73 \( 1 + 8.85T + 73T^{2} \)
79 \( 1 + 3.21T + 79T^{2} \)
83 \( 1 - 5.07T + 83T^{2} \)
89 \( 1 - 9.29T + 89T^{2} \)
97 \( 1 + 9.48T + 97T^{2} \)
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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

−8.110511649280919022850901135053, −7.26719810172754407210846294426, −6.50674767295582904490498543844, −6.13130693211787784550384823264, −5.05401721014930468928972435714, −4.29327286451243797158440685970, −3.61592553540299387490775940999, −3.12853154319458095474387536186, −1.79301271320798307341508595704, −1.19942123326799244765080431792, 1.19942123326799244765080431792, 1.79301271320798307341508595704, 3.12853154319458095474387536186, 3.61592553540299387490775940999, 4.29327286451243797158440685970, 5.05401721014930468928972435714, 6.13130693211787784550384823264, 6.50674767295582904490498543844, 7.26719810172754407210846294426, 8.110511649280919022850901135053

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