↑

Properties

Label 2-1134-21.5-c1-0-17
Degree $2$
Conductor $1134$
Sign $0.939 + 0.341i$
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 + (2.08 − 3.61i)5-s + (1.94 + 1.79i)7-s − 0.999i·8-s + (−3.61 + 2.08i)10-s + (0.0814 − 0.0470i)11-s + 3.79i·13-s + (−0.786 − 2.52i)14-s + (−0.5 + 0.866i)16-s + (3.82 + 6.62i)17-s + (6.77 + 3.91i)19-s + 4.17·20-s − 0.0940·22-s + (3.31 + 1.91i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.932 − 1.61i)5-s + (0.734 + 0.678i)7-s − 0.353i·8-s + (−1.14 + 0.659i)10-s + (0.0245 − 0.0141i)11-s + 1.05i·13-s + (−0.210 − 0.675i)14-s + (−0.125 + 0.216i)16-s + (0.928 + 1.60i)17-s + (1.55 + 0.897i)19-s + 0.932·20-s − 0.0200·22-s + (0.691 + 0.399i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.939 + 0.341i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (971, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.939 + 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.650298421\)
\(L(\frac12)\) \(\approx\) \(1.650298421\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.94 - 1.79i)T \)
good5 \( 1 + (-2.08 + 3.61i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.0814 + 0.0470i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.79iT - 13T^{2} \)
17 \( 1 + (-3.82 - 6.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.77 - 3.91i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.31 - 1.91i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.03iT - 29T^{2} \)
31 \( 1 + (1.45 - 0.840i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.18 - 3.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 5.12T + 43T^{2} \)
47 \( 1 + (-2.54 + 4.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.05 - 3.49i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.21 + 9.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.69 - 1.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.76 + 6.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.61iT - 71T^{2} \)
73 \( 1 + (0.129 - 0.0750i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.48 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.58T + 83T^{2} \)
89 \( 1 + (-1.37 + 2.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.06iT - 97T^{2} \)
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

−9.602000838490566521458571011877, −9.003197630448842726834441240798, −8.384075207101802073291112992965, −7.68480251861846463209387317737, −6.24368163561575498805585068488, −5.44114931494427170288026891710, −4.75600667441920953642114542842, −3.45489365333909983716146771451, −1.66406004004760020350547973429, −1.50371500428630189031356078500, 1.04248762407145452981947744609, 2.57640672821323983001539520944, 3.27143188047518975523673082930, 5.08141279182572396526637195280, 5.65554572276798262706671974743, 6.90190776865642713013894094221, 7.25328471576681120392707540177, 7.949788380483102397665924929050, 9.310293406586615226062504975924, 9.861180365676616202309277077855

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