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Properties

Label 2-273-91.24-c1-0-8
Degree $2$
Conductor $273$
Sign $0.799 - 0.601i$
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  + (−1.50 + 0.402i)2-s + i·3-s + (0.363 − 0.209i)4-s + (2.78 + 0.745i)5-s + (−0.402 − 1.50i)6-s + (0.599 − 2.57i)7-s + (1.73 − 1.73i)8-s − 9-s − 4.48·10-s + (3.87 − 3.87i)11-s + (0.209 + 0.363i)12-s + (0.662 + 3.54i)13-s + (0.136 + 4.11i)14-s + (−0.745 + 2.78i)15-s + (−2.33 + 4.03i)16-s + (−2.18 − 3.77i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.284i)2-s + 0.577i·3-s + (0.181 − 0.104i)4-s + (1.24 + 0.333i)5-s + (−0.164 − 0.613i)6-s + (0.226 − 0.973i)7-s + (0.614 − 0.614i)8-s − 0.333·9-s − 1.41·10-s + (1.16 − 1.16i)11-s + (0.0605 + 0.104i)12-s + (0.183 + 0.982i)13-s + (0.0363 + 1.09i)14-s + (−0.192 + 0.718i)15-s + (−0.582 + 1.00i)16-s + (−0.529 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.799 - 0.601i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.799 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.875469 + 0.292575i\)
\(L(\frac12)\) \(\approx\) \(0.875469 + 0.292575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.599 + 2.57i)T \)
13 \( 1 + (-0.662 - 3.54i)T \)
good2 \( 1 + (1.50 - 0.402i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-2.78 - 0.745i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.87 + 3.87i)T - 11iT^{2} \)
17 \( 1 + (2.18 + 3.77i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.02 - 1.02i)T - 19iT^{2} \)
23 \( 1 + (-7.25 - 4.18i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.882 - 1.52i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.206 - 0.770i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.41 - 5.27i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (3.88 + 1.03i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.58 + 3.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.28 - 8.52i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.139 + 0.241i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.89 + 7.05i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 4.74iT - 61T^{2} \)
67 \( 1 + (1.37 + 1.37i)T + 67iT^{2} \)
71 \( 1 + (-11.3 + 3.04i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (13.9 - 3.72i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.431 + 0.746i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.29 - 4.29i)T - 83iT^{2} \)
89 \( 1 + (-5.75 + 1.54i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.575 - 2.14i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.44809169719175504811067283372, −10.84024957399953244326511961040, −9.794029353042641095253681817287, −9.276642245492540727615087647740, −8.494103943098913688834185871488, −7.02464957842174156479000992544, −6.38537410901659991812228399620, −4.80091996337703458551259945946, −3.51139750272975775689836237222, −1.35048947078459026554450622433, 1.43820964217149893001803735887, 2.34759914271770757127921675132, 4.81174197281139875706887536506, 5.88259056415272385106413652217, 6.95858622744082877503719965176, 8.400266706020951199394666650859, 8.939053302252415953637504953725, 9.733090314425954743840917870194, 10.61689803446230937066085522238, 11.72156906817389928096639286422

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