Properties

Label 2-272-1.1-c9-0-4
Degree $2$
Conductor $272$
Sign $1$
Analytic cond. $140.089$
Root an. cond. $11.8359$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 177.·3-s − 1.62e3·5-s + 1.83e3·7-s + 1.18e4·9-s + 3.17e4·11-s − 1.32e5·13-s + 2.87e5·15-s − 8.35e4·17-s + 1.60e3·19-s − 3.25e5·21-s − 2.32e4·23-s + 6.71e5·25-s + 1.39e6·27-s + 3.73e6·29-s − 8.91e6·31-s − 5.63e6·33-s − 2.97e6·35-s − 1.20e7·37-s + 2.34e7·39-s − 1.26e7·41-s − 2.86e7·43-s − 1.91e7·45-s + 7.17e6·47-s − 3.69e7·49-s + 1.48e7·51-s − 5.96e7·53-s − 5.14e7·55-s + ⋯
L(s)  = 1  − 1.26·3-s − 1.15·5-s + 0.288·7-s + 0.599·9-s + 0.654·11-s − 1.28·13-s + 1.46·15-s − 0.242·17-s + 0.00282·19-s − 0.365·21-s − 0.0173·23-s + 0.343·25-s + 0.506·27-s + 0.980·29-s − 1.73·31-s − 0.827·33-s − 0.334·35-s − 1.05·37-s + 1.62·39-s − 0.700·41-s − 1.27·43-s − 0.695·45-s + 0.214·47-s − 0.916·49-s + 0.306·51-s − 1.03·53-s − 0.758·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(272\)    =    \(2^{4} \cdot 17\)
Sign: $1$
Analytic conductor: \(140.089\)
Root analytic conductor: \(11.8359\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{272} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 272,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.1675242320\)
\(L(\frac12)\) \(\approx\) \(0.1675242320\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + 8.35e4T \)
good3 \( 1 + 177.T + 1.96e4T^{2} \)
5 \( 1 + 1.62e3T + 1.95e6T^{2} \)
7 \( 1 - 1.83e3T + 4.03e7T^{2} \)
11 \( 1 - 3.17e4T + 2.35e9T^{2} \)
13 \( 1 + 1.32e5T + 1.06e10T^{2} \)
19 \( 1 - 1.60e3T + 3.22e11T^{2} \)
23 \( 1 + 2.32e4T + 1.80e12T^{2} \)
29 \( 1 - 3.73e6T + 1.45e13T^{2} \)
31 \( 1 + 8.91e6T + 2.64e13T^{2} \)
37 \( 1 + 1.20e7T + 1.29e14T^{2} \)
41 \( 1 + 1.26e7T + 3.27e14T^{2} \)
43 \( 1 + 2.86e7T + 5.02e14T^{2} \)
47 \( 1 - 7.17e6T + 1.11e15T^{2} \)
53 \( 1 + 5.96e7T + 3.29e15T^{2} \)
59 \( 1 + 1.85e8T + 8.66e15T^{2} \)
61 \( 1 + 2.00e8T + 1.16e16T^{2} \)
67 \( 1 - 1.27e8T + 2.72e16T^{2} \)
71 \( 1 - 3.27e8T + 4.58e16T^{2} \)
73 \( 1 + 1.48e8T + 5.88e16T^{2} \)
79 \( 1 - 2.58e8T + 1.19e17T^{2} \)
83 \( 1 + 3.45e8T + 1.86e17T^{2} \)
89 \( 1 - 4.03e8T + 3.50e17T^{2} \)
97 \( 1 + 9.89e8T + 7.60e17T^{2} \)
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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

−10.65802253706128520533359989553, −9.472273545305115502850072889136, −8.255104922991919870508170211493, −7.26626191912924290109569638044, −6.46167702994012844521563218312, −5.18003978440908109615766464440, −4.51529846814718957394177994208, −3.29696398692107505753022360022, −1.60904669376382880009188016657, −0.19785623225894636513856731954, 0.19785623225894636513856731954, 1.60904669376382880009188016657, 3.29696398692107505753022360022, 4.51529846814718957394177994208, 5.18003978440908109615766464440, 6.46167702994012844521563218312, 7.26626191912924290109569638044, 8.255104922991919870508170211493, 9.472273545305115502850072889136, 10.65802253706128520533359989553

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