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Properties

Label 2-273-273.152-c1-0-20
Degree $2$
Conductor $273$
Sign $0.964 - 0.263i$
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  + (0.730 − 0.421i)2-s + (1.64 + 0.548i)3-s + (−0.644 + 1.11i)4-s + (1.06 − 1.84i)5-s + (1.43 − 0.292i)6-s + (−0.110 + 2.64i)7-s + 2.77i·8-s + (2.39 + 1.80i)9-s − 1.79i·10-s − 0.987i·11-s + (−1.67 + 1.48i)12-s + (−3.08 − 1.86i)13-s + (1.03 + 1.97i)14-s + (2.75 − 2.44i)15-s + (−0.120 − 0.207i)16-s + (0.920 − 1.59i)17-s + ⋯
L(s)  = 1  + (0.516 − 0.298i)2-s + (0.948 + 0.316i)3-s + (−0.322 + 0.558i)4-s + (0.475 − 0.823i)5-s + (0.584 − 0.119i)6-s + (−0.0418 + 0.999i)7-s + 0.980i·8-s + (0.799 + 0.600i)9-s − 0.566i·10-s − 0.297i·11-s + (−0.482 + 0.427i)12-s + (−0.855 − 0.518i)13-s + (0.276 + 0.528i)14-s + (0.711 − 0.630i)15-s + (−0.0300 − 0.0519i)16-s + (0.223 − 0.386i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.964 - 0.263i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.964 - 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07571 + 0.278521i\)
\(L(\frac12)\) \(\approx\) \(2.07571 + 0.278521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.64 - 0.548i)T \)
7 \( 1 + (0.110 - 2.64i)T \)
13 \( 1 + (3.08 + 1.86i)T \)
good2 \( 1 + (-0.730 + 0.421i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.06 + 1.84i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 0.987iT - 11T^{2} \)
17 \( 1 + (-0.920 + 1.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.15iT - 19T^{2} \)
23 \( 1 + (-1.40 + 0.813i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.488 + 0.281i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.23 - 2.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.64 + 2.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.93 - 8.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.214 + 0.370i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.40 + 11.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.59 - 3.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.96 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 2.21iT - 61T^{2} \)
67 \( 1 + 2.45T + 67T^{2} \)
71 \( 1 + (2.48 - 1.43i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.31 - 4.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.27 - 12.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 + (-2.59 - 4.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.172 + 0.0997i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.27288348230292050853334792734, −11.18301683120609152032353604443, −9.722973593579165108471532037511, −8.966980511147041150585966430025, −8.457038237200807447409417074166, −7.24939958026483537523748757454, −5.29846255886161067516774100438, −4.80056019333574476118577385620, −3.24679218885964748384640152275, −2.32968379121440946808862780320, 1.74934189235310923784400843990, 3.45222624413740097614720939590, 4.45007454319730476990577601962, 5.99648409261601649550295985497, 6.95276550850683569327740636106, 7.65902769651816379286854815087, 9.149716923017155115852330229611, 10.07981492430190249178083012448, 10.46748301225568618507249223459, 12.20127337641693022209922123791

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