Properties

Label 2-6018-1.1-c1-0-7
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 − 3.00·5-s − 6-s − 0.900·7-s + 8-s + 9-s − 3.00·10-s − 6.30·11-s − 12-s − 3.45·13-s − 0.900·14-s + 3.00·15-s + 16-s + 17-s + 18-s + 6.45·19-s − 3.00·20-s + 0.900·21-s − 6.30·22-s − 2.88·23-s − 24-s + 4.05·25-s − 3.45·26-s − 27-s − 0.900·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.34·5-s − 0.408·6-s − 0.340·7-s + 0.353·8-s + 0.333·9-s − 0.951·10-s − 1.90·11-s − 0.288·12-s − 0.957·13-s − 0.240·14-s + 0.776·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.48·19-s − 0.672·20-s + 0.196·21-s − 1.34·22-s − 0.601·23-s − 0.204·24-s + 0.810·25-s − 0.676·26-s − 0.192·27-s − 0.170·28-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

\(L(1)\) \(\approx\) \(0.7781387341\)
\(L(\frac12)\) \(\approx\) \(0.7781387341\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.00T + 5T^{2} \)
7 \( 1 + 0.900T + 7T^{2} \)
11 \( 1 + 6.30T + 11T^{2} \)
13 \( 1 + 3.45T + 13T^{2} \)
19 \( 1 - 6.45T + 19T^{2} \)
23 \( 1 + 2.88T + 23T^{2} \)
29 \( 1 - 2.91T + 29T^{2} \)
31 \( 1 + 1.87T + 31T^{2} \)
37 \( 1 + 8.38T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + 4.63T + 43T^{2} \)
47 \( 1 + 9.49T + 47T^{2} \)
53 \( 1 - 6.66T + 53T^{2} \)
61 \( 1 - 1.98T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 + 9.78T + 71T^{2} \)
73 \( 1 + 4.23T + 73T^{2} \)
79 \( 1 - 5.62T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 - 9.46T + 89T^{2} \)
97 \( 1 - 7.99T + 97T^{2} \)
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

−7.906134638400736669436273413213, −7.28581212664282056704217303279, −6.80569110294145390810545442185, −5.68469679930362138374720409151, −5.03205800248013020627532942240, −4.74716062166608934575161857844, −3.47538087822207871716415768551, −3.18747777326527519009384079600, −2.01898313804184787912121790401, −0.40318287333234964180493139396, 0.40318287333234964180493139396, 2.01898313804184787912121790401, 3.18747777326527519009384079600, 3.47538087822207871716415768551, 4.74716062166608934575161857844, 5.03205800248013020627532942240, 5.68469679930362138374720409151, 6.80569110294145390810545442185, 7.28581212664282056704217303279, 7.906134638400736669436273413213

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