Properties

Label 2-6027-1.1-c1-0-258
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.07·2-s − 3-s + 2.31·4-s + 2.30·5-s − 2.07·6-s + 0.652·8-s + 9-s + 4.77·10-s − 2.24·11-s − 2.31·12-s − 4.05·13-s − 2.30·15-s − 3.27·16-s − 1.22·17-s + 2.07·18-s − 4.41·19-s + 5.32·20-s − 4.65·22-s + 6.56·23-s − 0.652·24-s + 0.291·25-s − 8.42·26-s − 27-s + 3.41·29-s − 4.77·30-s + 1.07·31-s − 8.10·32-s + ⋯
L(s)  = 1  + 1.46·2-s − 0.577·3-s + 1.15·4-s + 1.02·5-s − 0.847·6-s + 0.230·8-s + 0.333·9-s + 1.51·10-s − 0.675·11-s − 0.668·12-s − 1.12·13-s − 0.593·15-s − 0.818·16-s − 0.297·17-s + 0.489·18-s − 1.01·19-s + 1.19·20-s − 0.992·22-s + 1.36·23-s − 0.133·24-s + 0.0583·25-s − 1.65·26-s − 0.192·27-s + 0.634·29-s − 0.872·30-s + 0.193·31-s − 1.43·32-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: \(1\)
Selberg data: \((2,\ 6027,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.07T + 2T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 + 1.22T + 17T^{2} \)
19 \( 1 + 4.41T + 19T^{2} \)
23 \( 1 - 6.56T + 23T^{2} \)
29 \( 1 - 3.41T + 29T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 1.63T + 37T^{2} \)
43 \( 1 + 4.67T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 + 9.90T + 59T^{2} \)
61 \( 1 + 5.18T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 + 5.24T + 71T^{2} \)
73 \( 1 + 5.69T + 73T^{2} \)
79 \( 1 - 3.55T + 79T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 + 5.51T + 89T^{2} \)
97 \( 1 - 3.88T + 97T^{2} \)
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

−7.34061872983485789864177625447, −6.60211916190965815339510355381, −6.21025310836805875595138022118, −5.34456455049939107598595961792, −4.93537916161566226640258214747, −4.41283235784734950564402366594, −3.21126709897944034859692027498, −2.54085566634644347642758767544, −1.72611892031368568545852314446, 0, 1.72611892031368568545852314446, 2.54085566634644347642758767544, 3.21126709897944034859692027498, 4.41283235784734950564402366594, 4.93537916161566226640258214747, 5.34456455049939107598595961792, 6.21025310836805875595138022118, 6.60211916190965815339510355381, 7.34061872983485789864177625447

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