Properties

Label 2-273-91.41-c1-0-10
Degree $2$
Conductor $273$
Sign $0.893 - 0.448i$
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  + (1.34 + 0.359i)2-s + (0.866 − 0.5i)3-s + (−0.0625 − 0.0361i)4-s + (2.52 + 2.52i)5-s + (1.34 − 0.359i)6-s + (−0.324 + 2.62i)7-s + (−2.03 − 2.03i)8-s + (0.499 − 0.866i)9-s + (2.48 + 4.29i)10-s + (0.0529 − 0.197i)11-s − 0.0722·12-s + (−2.07 − 2.94i)13-s + (−1.37 + 3.40i)14-s + (3.45 + 0.925i)15-s + (−1.92 − 3.33i)16-s + (1.13 − 1.97i)17-s + ⋯
L(s)  = 1  + (0.948 + 0.254i)2-s + (0.499 − 0.288i)3-s + (−0.0312 − 0.0180i)4-s + (1.13 + 1.13i)5-s + (0.547 − 0.146i)6-s + (−0.122 + 0.992i)7-s + (−0.719 − 0.719i)8-s + (0.166 − 0.288i)9-s + (0.785 + 1.35i)10-s + (0.0159 − 0.0596i)11-s − 0.0208·12-s + (−0.575 − 0.817i)13-s + (−0.368 + 0.910i)14-s + (0.891 + 0.238i)15-s + (−0.481 − 0.833i)16-s + (0.275 − 0.477i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.893 - 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27855 + 0.539251i\)
\(L(\frac12)\) \(\approx\) \(2.27855 + 0.539251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.324 - 2.62i)T \)
13 \( 1 + (2.07 + 2.94i)T \)
good2 \( 1 + (-1.34 - 0.359i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-2.52 - 2.52i)T + 5iT^{2} \)
11 \( 1 + (-0.0529 + 0.197i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.13 + 1.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.50 - 0.402i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.59 + 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.75 + 8.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.75 - 2.75i)T + 31iT^{2} \)
37 \( 1 + (-1.17 + 4.39i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.36 - 5.07i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.76 - 1.01i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.42 - 9.42i)T - 47iT^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + (0.510 + 1.90i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.850 + 0.491i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-15.1 - 4.06i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.00355 - 0.0132i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.24 + 2.24i)T - 73iT^{2} \)
79 \( 1 - 12.7T + 79T^{2} \)
83 \( 1 + (-3.37 - 3.37i)T + 83iT^{2} \)
89 \( 1 + (-11.8 - 3.17i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.85 - 2.10i)T + (84.0 - 48.5i)T^{2} \)
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

−12.40608677687334767515654134233, −11.11877823272555924099541450117, −9.778338440523550692507433117091, −9.437846386983114891218330448053, −7.952879639390951577280101890318, −6.60245464267381376057084877433, −5.99791411966259150190154994648, −5.00689574279928895614886844274, −3.22852604147587164696991603510, −2.42598123481108873654723019830, 1.86060728249593175667926020945, 3.51647496624219865138392846798, 4.61200679757052132230147602224, 5.28538822757399693500962447995, 6.65820449830495983235974926420, 8.159124954828790374741479522205, 9.156814244000897337735503121793, 9.749992337042269007667594740177, 10.94210249008373857439727639598, 12.23951335864222348909601131541

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