↑

Properties

Label 2-273-21.5-c1-0-3
Degree $2$
Conductor $273$
Sign $-0.316 - 0.948i$
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  + (−0.987 − 0.570i)2-s + (1.12 + 1.31i)3-s + (−0.350 − 0.606i)4-s + (−0.578 + 1.00i)5-s + (−0.358 − 1.94i)6-s + (−2.55 + 0.695i)7-s + 3.07i·8-s + (−0.472 + 2.96i)9-s + (1.14 − 0.660i)10-s + (−2.88 + 1.66i)11-s + (0.405 − 1.14i)12-s − i·13-s + (2.91 + 0.768i)14-s + (−1.97 + 0.364i)15-s + (1.05 − 1.82i)16-s + (2.80 + 4.86i)17-s + ⋯
L(s)  = 1  + (−0.698 − 0.403i)2-s + (0.649 + 0.760i)3-s + (−0.175 − 0.303i)4-s + (−0.258 + 0.448i)5-s + (−0.146 − 0.792i)6-s + (−0.964 + 0.262i)7-s + 1.08i·8-s + (−0.157 + 0.987i)9-s + (0.361 − 0.208i)10-s + (−0.869 + 0.501i)11-s + (0.117 − 0.329i)12-s − 0.277i·13-s + (0.779 + 0.205i)14-s + (−0.509 + 0.0940i)15-s + (0.263 − 0.456i)16-s + (0.681 + 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.316 - 0.948i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.350002 + 0.485741i\)
\(L(\frac12)\) \(\approx\) \(0.350002 + 0.485741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.12 - 1.31i)T \)
7 \( 1 + (2.55 - 0.695i)T \)
13 \( 1 + iT \)
good2 \( 1 + (0.987 + 0.570i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.578 - 1.00i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.88 - 1.66i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.80 - 4.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.600 + 0.346i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.42 + 3.70i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.13iT - 29T^{2} \)
31 \( 1 + (-5.29 + 3.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.98 - 8.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.92T + 41T^{2} \)
43 \( 1 - 12.1T + 43T^{2} \)
47 \( 1 + (1.34 - 2.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.81 + 5.66i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.77 + 4.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.38 + 5.42i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.342 + 0.594i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.968iT - 71T^{2} \)
73 \( 1 + (-1.05 + 0.606i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.42 + 9.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (1.09 - 1.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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

−12.04220349053419736117850784077, −10.56826894189466439248221716149, −10.35983334536921663004285513567, −9.543847563475698652447153298127, −8.525699526390831303312142925378, −7.77152707043230309043998516082, −6.14542369950665084822299070986, −4.93554943448362876524178278862, −3.48283599209453319953970559781, −2.31196193781970777026822470994, 0.51918118501064487129189394723, 2.86802754810799996545762844022, 4.01233157958342723169621627076, 5.90566338617029361218102452733, 7.10977867764089808352867785353, 7.74844164871421798620764800758, 8.623032096695869919583976336785, 9.413759652239658063772962821462, 10.28233138709795377497054369513, 12.06557595620978349219199929906

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