Properties

Label 2-273-273.44-c1-0-27
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

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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: $\chi_{273} (44, \cdot )$
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 \( 1 + (-12.3 + 12.3i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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