Properties

Label 2-6027-1.1-c1-0-56
Degree $2$
Conductor $6027$
Sign $1$
Analytic cond. $48.1258$
Root an. cond. $6.93727$
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  + 0.712·2-s − 3-s − 1.49·4-s + 0.415·5-s − 0.712·6-s − 2.48·8-s + 9-s + 0.295·10-s + 2.75·11-s + 1.49·12-s + 4.20·13-s − 0.415·15-s + 1.21·16-s − 4.39·17-s + 0.712·18-s + 1.26·19-s − 0.619·20-s + 1.96·22-s + 4.53·23-s + 2.48·24-s − 4.82·25-s + 2.99·26-s − 27-s − 5.42·29-s − 0.295·30-s + 8.86·31-s + 5.84·32-s + ⋯
L(s)  = 1  + 0.503·2-s − 0.577·3-s − 0.746·4-s + 0.185·5-s − 0.290·6-s − 0.879·8-s + 0.333·9-s + 0.0935·10-s + 0.831·11-s + 0.430·12-s + 1.16·13-s − 0.107·15-s + 0.302·16-s − 1.06·17-s + 0.167·18-s + 0.290·19-s − 0.138·20-s + 0.419·22-s + 0.944·23-s + 0.508·24-s − 0.965·25-s + 0.587·26-s − 0.192·27-s − 1.00·29-s − 0.0540·30-s + 1.59·31-s + 1.03·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(48.1258\)
Root analytic conductor: \(6.93727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6027,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.691868093\)
\(L(\frac12)\) \(\approx\) \(1.691868093\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good2 \( 1 - 0.712T + 2T^{2} \)
5 \( 1 - 0.415T + 5T^{2} \)
11 \( 1 - 2.75T + 11T^{2} \)
13 \( 1 - 4.20T + 13T^{2} \)
17 \( 1 + 4.39T + 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 - 4.53T + 23T^{2} \)
29 \( 1 + 5.42T + 29T^{2} \)
31 \( 1 - 8.86T + 31T^{2} \)
37 \( 1 + 2.60T + 37T^{2} \)
43 \( 1 - 4.38T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 - 1.43T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 1.10T + 67T^{2} \)
71 \( 1 - 2.83T + 71T^{2} \)
73 \( 1 - 3.91T + 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + 4.04T + 83T^{2} \)
89 \( 1 + 1.02T + 89T^{2} \)
97 \( 1 + 7.88T + 97T^{2} \)
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

−8.290006478148765273662498347733, −7.13902207359284490745766444561, −6.42839449979036129467321103269, −5.92335976294429129899919103730, −5.20093027855524075889259814677, −4.39942416327964445453035008491, −3.88602213407417124552105498037, −3.04249942390932765773627912019, −1.71087689913734228786972634891, −0.68072142889546036272065040892, 0.68072142889546036272065040892, 1.71087689913734228786972634891, 3.04249942390932765773627912019, 3.88602213407417124552105498037, 4.39942416327964445453035008491, 5.20093027855524075889259814677, 5.92335976294429129899919103730, 6.42839449979036129467321103269, 7.13902207359284490745766444561, 8.290006478148765273662498347733

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