↑

Properties

Label 2-1134-21.17-c1-0-30
Degree $2$
Conductor $1134$
Sign $-0.998 + 0.0503i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.77 − 3.07i)5-s + (−1.14 − 2.38i)7-s − 0.999i·8-s + (−3.07 − 1.77i)10-s + (2.61 + 1.51i)11-s − 1.02i·13-s + (−2.18 − 1.49i)14-s + (−0.5 − 0.866i)16-s + (0.809 − 1.40i)17-s + (−7.12 + 4.11i)19-s − 3.55·20-s + 3.02·22-s + (2.90 − 1.67i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.794 − 1.37i)5-s + (−0.432 − 0.901i)7-s − 0.353i·8-s + (−0.972 − 0.561i)10-s + (0.789 + 0.455i)11-s − 0.284i·13-s + (−0.583 − 0.399i)14-s + (−0.125 − 0.216i)16-s + (0.196 − 0.339i)17-s + (−1.63 + 0.943i)19-s − 0.794·20-s + 0.644·22-s + (0.606 − 0.350i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.998 + 0.0503i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.998 + 0.0503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.370208100\)
\(L(\frac12)\) \(\approx\) \(1.370208100\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (1.14 + 2.38i)T \)
good5 \( 1 + (1.77 + 3.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.61 - 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.02iT - 13T^{2} \)
17 \( 1 + (-0.809 + 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.12 - 4.11i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.90 + 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.27iT - 29T^{2} \)
31 \( 1 + (5.18 + 2.99i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.92 - 5.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.0944T + 41T^{2} \)
43 \( 1 + 6.11T + 43T^{2} \)
47 \( 1 + (2.57 + 4.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.76 - 1.59i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.42 - 7.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.06 + 2.34i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.187 + 0.325i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + (-1.13 - 0.655i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.462 + 0.800i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-2.35 - 4.07i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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

−9.497158452513647882205196626062, −8.592119874574877613215019676028, −7.80052644174942903183255205882, −6.84617892990122768045655811837, −5.92259514487826653760355354143, −4.69719171280578256178349702254, −4.23968745733887038652035292169, −3.44185607645458336280931230581, −1.72565590767323524190256901179, −0.47934717941902692032401166472, 2.27488527976684735306995276844, 3.26861992150094994934443014540, 3.90208782877893535621146249965, 5.13513939196933212947576021223, 6.35441626911187610700910596974, 6.62251444929706538202168563717, 7.50088569592476679333786516325, 8.570868847991390855192025075721, 9.178170932352844917232698600369, 10.47930182092173912349124711915

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