Properties

Label 2-177-1.1-c11-0-1
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 83.3·2-s − 243·3-s + 4.90e3·4-s − 4.95e3·5-s + 2.02e4·6-s + 1.43e4·7-s − 2.38e5·8-s + 5.90e4·9-s + 4.13e5·10-s − 5.46e5·11-s − 1.19e6·12-s + 3.49e5·13-s − 1.19e6·14-s + 1.20e6·15-s + 9.83e6·16-s − 7.20e6·17-s − 4.92e6·18-s − 6.45e6·19-s − 2.43e7·20-s − 3.47e6·21-s + 4.55e7·22-s − 3.67e7·23-s + 5.79e7·24-s − 2.42e7·25-s − 2.91e7·26-s − 1.43e7·27-s + 7.01e7·28-s + ⋯
L(s)  = 1  − 1.84·2-s − 0.577·3-s + 2.39·4-s − 0.709·5-s + 1.06·6-s + 0.321·7-s − 2.57·8-s + 0.333·9-s + 1.30·10-s − 1.02·11-s − 1.38·12-s + 0.261·13-s − 0.592·14-s + 0.409·15-s + 2.34·16-s − 1.23·17-s − 0.614·18-s − 0.597·19-s − 1.70·20-s − 0.185·21-s + 1.88·22-s − 1.18·23-s + 1.48·24-s − 0.496·25-s − 0.481·26-s − 0.192·27-s + 0.770·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.03177217403\)
\(L(\frac12)\) \(\approx\) \(0.03177217403\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 + 83.3T + 2.04e3T^{2} \)
5 \( 1 + 4.95e3T + 4.88e7T^{2} \)
7 \( 1 - 1.43e4T + 1.97e9T^{2} \)
11 \( 1 + 5.46e5T + 2.85e11T^{2} \)
13 \( 1 - 3.49e5T + 1.79e12T^{2} \)
17 \( 1 + 7.20e6T + 3.42e13T^{2} \)
19 \( 1 + 6.45e6T + 1.16e14T^{2} \)
23 \( 1 + 3.67e7T + 9.52e14T^{2} \)
29 \( 1 - 1.87e6T + 1.22e16T^{2} \)
31 \( 1 - 4.50e7T + 2.54e16T^{2} \)
37 \( 1 - 2.68e8T + 1.77e17T^{2} \)
41 \( 1 - 5.73e6T + 5.50e17T^{2} \)
43 \( 1 - 5.13e7T + 9.29e17T^{2} \)
47 \( 1 + 9.01e8T + 2.47e18T^{2} \)
53 \( 1 - 1.88e9T + 9.26e18T^{2} \)
61 \( 1 + 5.74e8T + 4.35e19T^{2} \)
67 \( 1 - 5.65e9T + 1.22e20T^{2} \)
71 \( 1 + 1.97e10T + 2.31e20T^{2} \)
73 \( 1 + 2.79e10T + 3.13e20T^{2} \)
79 \( 1 + 2.92e10T + 7.47e20T^{2} \)
83 \( 1 + 5.40e9T + 1.28e21T^{2} \)
89 \( 1 + 6.94e10T + 2.77e21T^{2} \)
97 \( 1 + 4.27e10T + 7.15e21T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.61342984158865008578927007165, −9.727761861638845646894489586661, −8.483715473096418483948335902190, −7.926910296297469480657295598129, −6.94930275234760610680856371099, −5.90863395600798758744927379750, −4.30461736691449405123966649607, −2.56868627585603782173395612448, −1.51904800349624438382167434832, −0.11044563945977581989846860035, 0.11044563945977581989846860035, 1.51904800349624438382167434832, 2.56868627585603782173395612448, 4.30461736691449405123966649607, 5.90863395600798758744927379750, 6.94930275234760610680856371099, 7.926910296297469480657295598129, 8.483715473096418483948335902190, 9.727761861638845646894489586661, 10.61342984158865008578927007165

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