Properties

Label 2-6018-1.1-c1-0-99
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 + 2.63·5-s − 6-s + 3.87·7-s − 8-s + 9-s − 2.63·10-s + 3.26·11-s + 12-s + 2.63·13-s − 3.87·14-s + 2.63·15-s + 16-s − 17-s − 18-s + 3.97·19-s + 2.63·20-s + 3.87·21-s − 3.26·22-s + 2.46·23-s − 24-s + 1.92·25-s − 2.63·26-s + 27-s + 3.87·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.17·5-s − 0.408·6-s + 1.46·7-s − 0.353·8-s + 0.333·9-s − 0.831·10-s + 0.983·11-s + 0.288·12-s + 0.729·13-s − 1.03·14-s + 0.679·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s + 0.911·19-s + 0.588·20-s + 0.846·21-s − 0.695·22-s + 0.513·23-s − 0.204·24-s + 0.384·25-s − 0.515·26-s + 0.192·27-s + 0.733·28-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

\(L(1)\) \(\approx\) \(3.330784830\)
\(L(\frac12)\) \(\approx\) \(3.330784830\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.63T + 5T^{2} \)
7 \( 1 - 3.87T + 7T^{2} \)
11 \( 1 - 3.26T + 11T^{2} \)
13 \( 1 - 2.63T + 13T^{2} \)
19 \( 1 - 3.97T + 19T^{2} \)
23 \( 1 - 2.46T + 23T^{2} \)
29 \( 1 - 1.70T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 6.70T + 41T^{2} \)
43 \( 1 - 0.921T + 43T^{2} \)
47 \( 1 + 5.70T + 47T^{2} \)
53 \( 1 - 8.03T + 53T^{2} \)
61 \( 1 + 4.73T + 61T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 - 3.29T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 9T + 97T^{2} \)
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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

−8.340365119259009982276595891550, −7.41609238469507455152887884809, −6.87304385681632248666548337548, −5.96711737475779556465354158217, −5.34635942423830407206664625857, −4.45466054222496036127383963286, −3.50075244519821392922104088434, −2.48142612283272561415117130439, −1.58273255296681098936169004989, −1.26123698876824587865549024318, 1.26123698876824587865549024318, 1.58273255296681098936169004989, 2.48142612283272561415117130439, 3.50075244519821392922104088434, 4.45466054222496036127383963286, 5.34635942423830407206664625857, 5.96711737475779556465354158217, 6.87304385681632248666548337548, 7.41609238469507455152887884809, 8.340365119259009982276595891550

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