Properties

Label 2-273-1.1-c1-0-9
Degree $2$
Conductor $273$
Sign $1$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.61·2-s + 3-s + 4.81·4-s − 3.81·5-s + 2.61·6-s + 7-s + 7.34·8-s + 9-s − 9.95·10-s − 4.73·11-s + 4.81·12-s + 13-s + 2.61·14-s − 3.81·15-s + 9.55·16-s − 5.22·17-s + 2.61·18-s + 2.92·19-s − 18.3·20-s + 21-s − 12.3·22-s + 3.33·23-s + 7.34·24-s + 9.55·25-s + 2.61·26-s + 27-s + 4.81·28-s + ⋯
L(s)  = 1  + 1.84·2-s + 0.577·3-s + 2.40·4-s − 1.70·5-s + 1.06·6-s + 0.377·7-s + 2.59·8-s + 0.333·9-s − 3.14·10-s − 1.42·11-s + 1.38·12-s + 0.277·13-s + 0.697·14-s − 0.984·15-s + 2.38·16-s − 1.26·17-s + 0.615·18-s + 0.670·19-s − 4.10·20-s + 0.218·21-s − 2.63·22-s + 0.694·23-s + 1.49·24-s + 1.91·25-s + 0.511·26-s + 0.192·27-s + 0.909·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(3.255660512\)
\(L(\frac12)\) \(\approx\) \(3.255660512\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 - 2.61T + 2T^{2} \)
5 \( 1 + 3.81T + 5T^{2} \)
11 \( 1 + 4.73T + 11T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 - 2.92T + 19T^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 + 0.922T + 29T^{2} \)
31 \( 1 + 7.51T + 31T^{2} \)
37 \( 1 - 0.154T + 37T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 + 6.55T + 43T^{2} \)
47 \( 1 - 9.03T + 47T^{2} \)
53 \( 1 - 8.55T + 53T^{2} \)
59 \( 1 - 3.95T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + 6.58T + 71T^{2} \)
73 \( 1 + 7.73T + 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + 1.40T + 83T^{2} \)
89 \( 1 + 1.96T + 89T^{2} \)
97 \( 1 + 2.11T + 97T^{2} \)
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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

−12.09501066929474667624338485190, −11.23939024306669450237144544528, −10.70243886733941868471052610459, −8.665553729300073867917416106964, −7.59812598639803922425882860881, −7.07590021453430569261566645838, −5.41369456039447429979351854216, −4.47083612524782670720743209297, −3.62387459756726331630318828744, −2.55677068828348872473276058447, 2.55677068828348872473276058447, 3.62387459756726331630318828744, 4.47083612524782670720743209297, 5.41369456039447429979351854216, 7.07590021453430569261566645838, 7.59812598639803922425882860881, 8.665553729300073867917416106964, 10.70243886733941868471052610459, 11.23939024306669450237144544528, 12.09501066929474667624338485190

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