Properties

Label 2-1368-1.1-c1-0-17
Degree $2$
Conductor $1368$
Sign $-1$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  − 0.319·5-s − 2.61·7-s − 4.31·11-s + 6.51·13-s + 4.19·17-s − 19-s + 0.639·23-s − 4.89·25-s − 7.87·29-s − 6·31-s + 0.837·35-s + 4·37-s − 12.5·41-s − 1.38·43-s − 7.55·47-s − 0.137·49-s − 13.1·53-s + 1.38·55-s − 13.0·59-s − 5.89·61-s − 2.08·65-s + 11.7·67-s + 11.7·71-s + 15.1·73-s + 11.3·77-s + 0.517·79-s − 11.8·83-s + ⋯
L(s)  = 1  − 0.142·5-s − 0.990·7-s − 1.30·11-s + 1.80·13-s + 1.01·17-s − 0.229·19-s + 0.133·23-s − 0.979·25-s − 1.46·29-s − 1.07·31-s + 0.141·35-s + 0.657·37-s − 1.95·41-s − 0.210·43-s − 1.10·47-s − 0.0196·49-s − 1.80·53-s + 0.186·55-s − 1.69·59-s − 0.755·61-s − 0.258·65-s + 1.43·67-s + 1.39·71-s + 1.77·73-s + 1.28·77-s + 0.0582·79-s − 1.30·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + T \)
good5 \( 1 + 0.319T + 5T^{2} \)
7 \( 1 + 2.61T + 7T^{2} \)
11 \( 1 + 4.31T + 11T^{2} \)
13 \( 1 - 6.51T + 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
23 \( 1 - 0.639T + 23T^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 + 1.38T + 43T^{2} \)
47 \( 1 + 7.55T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 5.89T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 15.1T + 73T^{2} \)
79 \( 1 - 0.517T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 - 3.87T + 89T^{2} \)
97 \( 1 - 11.2T + 97T^{2} \)
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

−9.331061914316843380690566436139, −8.214158399863934458841061598837, −7.76236100058198358688056494168, −6.60402266167481482864182587704, −5.90462987190852553162432134615, −5.12071139224204465003342190328, −3.66153756652950122022796613694, −3.25583784601214370055394952824, −1.73503719249279207763517610177, 0, 1.73503719249279207763517610177, 3.25583784601214370055394952824, 3.66153756652950122022796613694, 5.12071139224204465003342190328, 5.90462987190852553162432134615, 6.60402266167481482864182587704, 7.76236100058198358688056494168, 8.214158399863934458841061598837, 9.331061914316843380690566436139

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