↑

Properties

Label 2-1368-1368.1051-c0-0-0
Degree $2$
Conductor $1368$
Sign $0.188 - 0.982i$
Analytic cond. $0.682720$
Root an. cond. $0.826269$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)2-s + 3-s + (−0.939 + 0.342i)4-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + 9-s + 1.53·11-s + (−0.939 + 0.342i)12-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + (0.173 + 0.984i)18-s + (−0.939 + 0.342i)19-s + (0.266 + 1.50i)22-s + (−0.5 − 0.866i)24-s + (0.173 − 0.984i)25-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + 3-s + (−0.939 + 0.342i)4-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + 9-s + 1.53·11-s + (−0.939 + 0.342i)12-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + (0.173 + 0.984i)18-s + (−0.939 + 0.342i)19-s + (0.266 + 1.50i)22-s + (−0.5 − 0.866i)24-s + (0.173 − 0.984i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(0.682720\)
Root analytic conductor: \(0.826269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :0),\ 0.188 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.559702292\)
\(L(\frac12)\) \(\approx\) \(1.559702292\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 - T \)
19 \( 1 + (0.939 - 0.342i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - 1.53T + T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
97 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
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

−9.644588598169575563545602590437, −8.839251783025759231413925949064, −8.447611313288370694579297016318, −7.59626234976418489657337340151, −6.50705252398641487236156562425, −6.33090661288406209697110685151, −4.67394843388091085531201171298, −4.13177413080031401308217824498, −3.22134556471724146144083999877, −1.69916276353853756118770485195, 1.43845981839743621590915414958, 2.40723046801462471112230810726, 3.49145132306645658100101960287, 4.15604159493434402295917559932, 5.03280134109724917289783525895, 6.42330283254443637049414934348, 7.19383257656430740923727472997, 8.464115564058963581184869630547, 8.929169300938310182303125130055, 9.521618508322374557271220242217

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