Properties

Label 2-273-273.242-c1-0-2
Degree $2$
Conductor $273$
Sign $-0.996 - 0.0883i$
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  + (−2.41 − 0.646i)2-s + (0.872 + 1.49i)3-s + (3.67 + 2.11i)4-s + (−1.51 − 0.406i)5-s + (−1.13 − 4.17i)6-s + (−1.79 + 1.94i)7-s + (−3.95 − 3.95i)8-s + (−1.47 + 2.61i)9-s + (3.39 + 1.96i)10-s + (−1.16 + 0.312i)11-s + (0.0328 + 7.34i)12-s + (−1.67 − 3.19i)13-s + (5.57 − 3.54i)14-s + (−0.715 − 2.62i)15-s + (2.74 + 4.74i)16-s + (−0.0180 + 0.0312i)17-s + ⋯
L(s)  = 1  + (−1.70 − 0.457i)2-s + (0.503 + 0.863i)3-s + (1.83 + 1.05i)4-s + (−0.678 − 0.181i)5-s + (−0.464 − 1.70i)6-s + (−0.676 + 0.736i)7-s + (−1.39 − 1.39i)8-s + (−0.492 + 0.870i)9-s + (1.07 + 0.619i)10-s + (−0.351 + 0.0940i)11-s + (0.00949 + 2.11i)12-s + (−0.465 − 0.885i)13-s + (1.49 − 0.946i)14-s + (−0.184 − 0.677i)15-s + (0.685 + 1.18i)16-s + (−0.00437 + 0.00757i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.996 - 0.0883i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ -0.996 - 0.0883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00597389 + 0.134945i\)
\(L(\frac12)\) \(\approx\) \(0.00597389 + 0.134945i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.872 - 1.49i)T \)
7 \( 1 + (1.79 - 1.94i)T \)
13 \( 1 + (1.67 + 3.19i)T \)
good2 \( 1 + (2.41 + 0.646i)T + (1.73 + i)T^{2} \)
5 \( 1 + (1.51 + 0.406i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.16 - 0.312i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.0180 - 0.0312i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.354 + 1.32i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.87 + 4.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.4iT - 29T^{2} \)
31 \( 1 + (1.26 - 0.338i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (9.47 + 2.53i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.92 - 2.92i)T - 41iT^{2} \)
43 \( 1 - 4.27iT - 43T^{2} \)
47 \( 1 + (0.106 - 0.398i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (7.13 + 4.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.56 + 2.02i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.41 - 9.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.10 + 0.831i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.73 + 5.73i)T - 71iT^{2} \)
73 \( 1 + (-3.30 - 12.3i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.15 - 1.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.19 - 9.19i)T - 83iT^{2} \)
89 \( 1 + (1.95 - 7.31i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-12.3 - 12.3i)T + 97iT^{2} \)
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

−12.06117582914409201467898546109, −10.98741075896292239599257187650, −10.24038833462244127253994826803, −9.546367102612470075642097299626, −8.586644593618582116357748338624, −8.133815109401711714453393233781, −6.95023055787161743971361045168, −5.20937237699921499894411444228, −3.44222794444295657313004394806, −2.43530720256565477264529270610, 0.15554403712988962562152095479, 1.94677439021738132140602594129, 3.65706502210427749842520150033, 6.08224711428371688849776778697, 7.06569598619945472412559006618, 7.56781570265720254606044699939, 8.362981320438823941934266036931, 9.451801787407075103598399334758, 10.08589038608395653797999567998, 11.34359176016989604805511434652

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