↑

Properties

Label 2-273-91.88-c1-0-6
Degree $2$
Conductor $273$
Sign $-0.291 + 0.956i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
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  + (−2.11 − 1.22i)2-s − 3-s + (1.98 + 3.44i)4-s + (−3.36 + 1.94i)5-s + (2.11 + 1.22i)6-s + (−0.745 + 2.53i)7-s − 4.82i·8-s + 9-s + 9.50·10-s − 4.52i·11-s + (−1.98 − 3.44i)12-s + (−2.00 − 2.99i)13-s + (4.68 − 4.46i)14-s + (3.36 − 1.94i)15-s + (−1.92 + 3.32i)16-s + (3.16 + 5.47i)17-s + ⋯
L(s)  = 1  + (−1.49 − 0.864i)2-s − 0.577·3-s + (0.993 + 1.72i)4-s + (−1.50 + 0.869i)5-s + (0.864 + 0.498i)6-s + (−0.281 + 0.959i)7-s − 1.70i·8-s + 0.333·9-s + 3.00·10-s − 1.36i·11-s + (−0.573 − 0.993i)12-s + (−0.555 − 0.831i)13-s + (1.25 − 1.19i)14-s + (0.869 − 0.502i)15-s + (−0.480 + 0.831i)16-s + (0.766 + 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.291 + 0.956i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.291 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134130 - 0.181142i\)
\(L(\frac12)\) \(\approx\) \(0.134130 - 0.181142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.745 - 2.53i)T \)
13 \( 1 + (2.00 + 2.99i)T \)
good2 \( 1 + (2.11 + 1.22i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (3.36 - 1.94i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 4.52iT - 11T^{2} \)
17 \( 1 + (-3.16 - 5.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 1.94iT - 19T^{2} \)
23 \( 1 + (-1.33 + 2.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.99 + 3.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.77 - 1.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.60 + 1.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.70 + 2.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.41 + 2.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.60 + 0.926i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.173 - 0.300i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.25 + 3.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 0.247iT - 67T^{2} \)
71 \( 1 + (-9.28 - 5.36i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.89 + 3.98i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.35 + 11.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.782iT - 83T^{2} \)
89 \( 1 + (10.6 + 6.15i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.4 + 6.59i)T + (48.5 + 84.0i)T^{2} \)
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

−11.33145894350814746541444079970, −10.80322146692350562938144902650, −9.991983819865329155464180945165, −8.606140872214966034283333825444, −8.101530111863656434833615547912, −7.08575695952902375739889847511, −5.77349633092896193509997265227, −3.64872336751449029195898595325, −2.73541961286316977010119004806, −0.37986308041237505079965718261, 1.06553322001924790239848807702, 4.13871556463092396655716493142, 5.16077485618137811046663908465, 7.01688049454320366278716290738, 7.25294230597753786793468638282, 8.102935574064667726226534158432, 9.395577068136964271919550904695, 9.891362264696437549861113328013, 11.11809197192881384203274841116, 11.93357923065405882894062271035

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