Properties

Label 2-273-1.1-c1-0-8
Degree $2$
Conductor $273$
Sign $1$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 7-s + 9-s + 2·10-s − 2·11-s + 2·12-s − 13-s − 2·14-s + 15-s − 4·16-s + 2·18-s + 19-s + 2·20-s − 21-s − 4·22-s + 3·23-s − 4·25-s − 2·26-s + 27-s − 2·28-s − 5·29-s + 2·30-s + 9·31-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s − 0.603·11-s + 0.577·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s − 16-s + 0.471·18-s + 0.229·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.625·23-s − 4/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.928·29-s + 0.365·30-s + 1.61·31-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.852768318\)
\(L(\frac12)\) \(\approx\) \(2.852768318\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 15 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−12.28936108770676961423612040714, −11.22516944735448501879908089914, −9.997612005245311033735868727795, −9.153614989507917020414880699166, −7.85096152981510062691288793912, −6.66540549198067832227195282850, −5.65939043191698227214275856071, −4.64274960483946295855931167095, −3.41858543863816620367865257858, −2.38895973908794646217186104367, 2.38895973908794646217186104367, 3.41858543863816620367865257858, 4.64274960483946295855931167095, 5.65939043191698227214275856071, 6.66540549198067832227195282850, 7.85096152981510062691288793912, 9.153614989507917020414880699166, 9.997612005245311033735868727795, 11.22516944735448501879908089914, 12.28936108770676961423612040714

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