Properties

Label 2-273-1.1-c11-0-57
Degree $2$
Conductor $273$
Sign $-1$
Analytic cond. $209.757$
Root an. cond. $14.4830$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 88.0·2-s − 243·3-s + 5.70e3·4-s − 1.34e4·5-s + 2.14e4·6-s + 1.68e4·7-s − 3.22e5·8-s + 5.90e4·9-s + 1.18e6·10-s + 9.66e5·11-s − 1.38e6·12-s − 3.71e5·13-s − 1.48e6·14-s + 3.25e6·15-s + 1.67e7·16-s − 3.26e6·17-s − 5.20e6·18-s + 7.04e6·19-s − 7.65e7·20-s − 4.08e6·21-s − 8.51e7·22-s − 2.58e7·23-s + 7.83e7·24-s + 1.30e8·25-s + 3.27e7·26-s − 1.43e7·27-s + 9.59e7·28-s + ⋯
L(s)  = 1  − 1.94·2-s − 0.577·3-s + 2.78·4-s − 1.91·5-s + 1.12·6-s + 0.377·7-s − 3.47·8-s + 0.333·9-s + 3.73·10-s + 1.80·11-s − 1.60·12-s − 0.277·13-s − 0.735·14-s + 1.10·15-s + 3.98·16-s − 0.558·17-s − 0.648·18-s + 0.652·19-s − 5.34·20-s − 0.218·21-s − 3.52·22-s − 0.836·23-s + 2.00·24-s + 2.67·25-s + 0.539·26-s − 0.192·27-s + 1.05·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(209.757\)
Root analytic conductor: \(14.4830\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 273,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
7 \( 1 - 1.68e4T \)
13 \( 1 + 3.71e5T \)
good2 \( 1 + 88.0T + 2.04e3T^{2} \)
5 \( 1 + 1.34e4T + 4.88e7T^{2} \)
11 \( 1 - 9.66e5T + 2.85e11T^{2} \)
17 \( 1 + 3.26e6T + 3.42e13T^{2} \)
19 \( 1 - 7.04e6T + 1.16e14T^{2} \)
23 \( 1 + 2.58e7T + 9.52e14T^{2} \)
29 \( 1 + 9.40e7T + 1.22e16T^{2} \)
31 \( 1 - 1.08e8T + 2.54e16T^{2} \)
37 \( 1 + 5.39e8T + 1.77e17T^{2} \)
41 \( 1 - 6.21e8T + 5.50e17T^{2} \)
43 \( 1 - 2.48e8T + 9.29e17T^{2} \)
47 \( 1 - 1.09e8T + 2.47e18T^{2} \)
53 \( 1 - 3.34e8T + 9.26e18T^{2} \)
59 \( 1 + 7.67e9T + 3.01e19T^{2} \)
61 \( 1 + 1.30e9T + 4.35e19T^{2} \)
67 \( 1 + 1.72e10T + 1.22e20T^{2} \)
71 \( 1 + 2.59e10T + 2.31e20T^{2} \)
73 \( 1 - 9.41e9T + 3.13e20T^{2} \)
79 \( 1 + 1.58e10T + 7.47e20T^{2} \)
83 \( 1 - 1.74e10T + 1.28e21T^{2} \)
89 \( 1 - 3.74e10T + 2.77e21T^{2} \)
97 \( 1 + 3.78e10T + 7.15e21T^{2} \)
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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

−9.287019433234962638610899969180, −8.626121023746689997532183368920, −7.62988128121361934575460360130, −7.13861180642953656766950600691, −6.17716649551105149785976090824, −4.34397542989179058898335040290, −3.29245424626590567049222695598, −1.67904639975996568713849948881, −0.795376565750484546912950739255, 0, 0.795376565750484546912950739255, 1.67904639975996568713849948881, 3.29245424626590567049222695598, 4.34397542989179058898335040290, 6.17716649551105149785976090824, 7.13861180642953656766950600691, 7.62988128121361934575460360130, 8.626121023746689997532183368920, 9.287019433234962638610899969180

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