Properties

Label 2-273-91.80-c1-0-11
Degree $2$
Conductor $273$
Sign $0.867 + 0.497i$
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  + (−0.0881 − 0.328i)2-s + i·3-s + (1.63 − 0.942i)4-s + (0.552 − 2.06i)5-s + (0.328 − 0.0881i)6-s + (2.31 + 1.28i)7-s + (−0.935 − 0.935i)8-s − 9-s − 0.727·10-s + (−0.152 − 0.152i)11-s + (0.942 + 1.63i)12-s + (−3.58 − 0.408i)13-s + (0.219 − 0.873i)14-s + (2.06 + 0.552i)15-s + (1.65 − 2.87i)16-s + (−2.64 − 4.58i)17-s + ⋯
L(s)  = 1  + (−0.0623 − 0.232i)2-s + 0.577i·3-s + (0.815 − 0.471i)4-s + (0.247 − 0.922i)5-s + (0.134 − 0.0359i)6-s + (0.873 + 0.486i)7-s + (−0.330 − 0.330i)8-s − 0.333·9-s − 0.230·10-s + (−0.0460 − 0.0460i)11-s + (0.271 + 0.471i)12-s + (−0.993 − 0.113i)13-s + (0.0586 − 0.233i)14-s + (0.532 + 0.142i)15-s + (0.414 − 0.718i)16-s + (−0.641 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.867 + 0.497i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.867 + 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50620 - 0.401226i\)
\(L(\frac12)\) \(\approx\) \(1.50620 - 0.401226i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-2.31 - 1.28i)T \)
13 \( 1 + (3.58 + 0.408i)T \)
good2 \( 1 + (0.0881 + 0.328i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.552 + 2.06i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.152 + 0.152i)T + 11iT^{2} \)
17 \( 1 + (2.64 + 4.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.54 - 4.54i)T + 19iT^{2} \)
23 \( 1 + (-5.58 - 3.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.21 - 2.10i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.54 - 0.681i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.73 - 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.28 - 8.53i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.95 + 2.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (12.3 + 3.31i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.97 + 3.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.8 + 3.16i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 2.81iT - 61T^{2} \)
67 \( 1 + (-1.19 + 1.19i)T - 67iT^{2} \)
71 \( 1 + (-0.653 - 2.44i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.00884 - 0.0329i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.28 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.72 - 7.72i)T + 83iT^{2} \)
89 \( 1 + (-2.16 - 8.08i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.90 - 2.11i)T + (84.0 - 48.5i)T^{2} \)
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

−11.71001080788018223090231043550, −11.00930021941161805897951134559, −9.813372574978192004824993962626, −9.251394423327180920565208303799, −8.061146669268871689193973361339, −6.88865459033664846011803745064, −5.18246870842428191484824922333, −5.13306108917601289262550580335, −3.02571641325742706579234198300, −1.53775833341907410607993740972, 1.99885742438578831955221150082, 3.11625718664740241919952376163, 4.90544621579746579227042069395, 6.41442065295025396554104313492, 7.09106948135411881425108448039, 7.75894778987312503898707445418, 8.882584263621120175818581299252, 10.44951381328856685967439409234, 11.06027487656995103929126884724, 11.85684605792764525845238815918

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