↑

Properties

Label 2-273-91.16-c1-0-10
Degree $2$
Conductor $273$
Sign $0.0879 + 0.996i$
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  − 2.71·2-s + (−0.5 + 0.866i)3-s + 5.37·4-s + (1.94 − 3.36i)5-s + (1.35 − 2.35i)6-s + (1.94 − 1.79i)7-s − 9.16·8-s + (−0.499 − 0.866i)9-s + (−5.27 + 9.14i)10-s + (0.815 − 1.41i)11-s + (−2.68 + 4.65i)12-s + (−3.59 + 0.202i)13-s + (−5.26 + 4.88i)14-s + (1.94 + 3.36i)15-s + 14.1·16-s − 4.19·17-s + ⋯
L(s)  = 1  − 1.92·2-s + (−0.288 + 0.499i)3-s + 2.68·4-s + (0.869 − 1.50i)5-s + (0.554 − 0.960i)6-s + (0.733 − 0.679i)7-s − 3.23·8-s + (−0.166 − 0.288i)9-s + (−1.66 + 2.89i)10-s + (0.245 − 0.426i)11-s + (−0.775 + 1.34i)12-s + (−0.998 + 0.0562i)13-s + (−1.40 + 1.30i)14-s + (0.501 + 0.869i)15-s + 3.53·16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0879 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0879 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.0879 + 0.996i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.0879 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391590 - 0.358555i\)
\(L(\frac12)\) \(\approx\) \(0.391590 - 0.358555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.94 + 1.79i)T \)
13 \( 1 + (3.59 - 0.202i)T \)
good2 \( 1 + 2.71T + 2T^{2} \)
5 \( 1 + (-1.94 + 3.36i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.815 + 1.41i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.19T + 17T^{2} \)
19 \( 1 + (0.847 + 1.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.791T + 23T^{2} \)
29 \( 1 + (0.242 + 0.419i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.915 + 1.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.689T + 37T^{2} \)
41 \( 1 + (-2.96 - 5.14i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.79 + 4.83i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.292 - 0.506i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.04 - 5.28i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.27T + 59T^{2} \)
61 \( 1 + (-4.54 - 7.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.500 - 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.93 + 13.7i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.92 + 5.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.643 + 1.11i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.18T + 83T^{2} \)
89 \( 1 + 4.40T + 89T^{2} \)
97 \( 1 + (-5.08 + 8.81i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.29776090770989280768147478332, −10.45227402425397680480058755671, −9.584493066301003352445073728264, −8.972894744934917913393027676833, −8.226124613672275307202731525988, −7.06351902093272105962352267483, −5.81322168233146513261594515745, −4.58616536891088233128751786597, −2.11543938985423384066450624162, −0.75541280836021356136935539788, 1.94556791580524943201476182951, 2.56615959682130796416494193966, 5.66534803539587680258799587704, 6.72046143899253528489873997923, 7.21631610834965623295991394909, 8.306447175233007959977876499984, 9.392206736880869210089257955266, 10.13926727860152026429832647852, 11.01156010388082800918675588230, 11.53985112755651622295917839604

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