Properties

Label 2-177-1.1-c11-0-4
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  − 44.5·2-s + 243·3-s − 66.0·4-s − 2.49e3·5-s − 1.08e4·6-s − 1.02e4·7-s + 9.41e4·8-s + 5.90e4·9-s + 1.10e5·10-s − 9.35e5·11-s − 1.60e4·12-s − 1.88e6·13-s + 4.54e5·14-s − 6.05e5·15-s − 4.05e6·16-s − 7.73e6·17-s − 2.62e6·18-s + 2.90e6·19-s + 1.64e5·20-s − 2.48e6·21-s + 4.16e7·22-s + 3.68e7·23-s + 2.28e7·24-s − 4.26e7·25-s + 8.37e7·26-s + 1.43e7·27-s + 6.74e5·28-s + ⋯
L(s)  = 1  − 0.983·2-s + 0.577·3-s − 0.0322·4-s − 0.356·5-s − 0.567·6-s − 0.229·7-s + 1.01·8-s + 0.333·9-s + 0.350·10-s − 1.75·11-s − 0.0186·12-s − 1.40·13-s + 0.226·14-s − 0.205·15-s − 0.966·16-s − 1.32·17-s − 0.327·18-s + 0.268·19-s + 0.0114·20-s − 0.132·21-s + 1.72·22-s + 1.19·23-s + 0.586·24-s − 0.872·25-s + 1.38·26-s + 0.192·27-s + 0.00741·28-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

\(L(6)\) \(\approx\) \(0.1284898606\)
\(L(\frac12)\) \(\approx\) \(0.1284898606\)
\(L(\frac{13}{2})\) 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 \( 1 + 44.5T + 2.04e3T^{2} \)
5 \( 1 + 2.49e3T + 4.88e7T^{2} \)
7 \( 1 + 1.02e4T + 1.97e9T^{2} \)
11 \( 1 + 9.35e5T + 2.85e11T^{2} \)
13 \( 1 + 1.88e6T + 1.79e12T^{2} \)
17 \( 1 + 7.73e6T + 3.42e13T^{2} \)
19 \( 1 - 2.90e6T + 1.16e14T^{2} \)
23 \( 1 - 3.68e7T + 9.52e14T^{2} \)
29 \( 1 + 1.85e8T + 1.22e16T^{2} \)
31 \( 1 + 9.30e7T + 2.54e16T^{2} \)
37 \( 1 + 2.96e8T + 1.77e17T^{2} \)
41 \( 1 - 3.54e8T + 5.50e17T^{2} \)
43 \( 1 - 1.17e9T + 9.29e17T^{2} \)
47 \( 1 + 3.82e8T + 2.47e18T^{2} \)
53 \( 1 + 1.18e9T + 9.26e18T^{2} \)
61 \( 1 - 3.90e9T + 4.35e19T^{2} \)
67 \( 1 + 9.49e9T + 1.22e20T^{2} \)
71 \( 1 + 2.28e10T + 2.31e20T^{2} \)
73 \( 1 + 1.82e10T + 3.13e20T^{2} \)
79 \( 1 + 2.69e10T + 7.47e20T^{2} \)
83 \( 1 + 5.51e10T + 1.28e21T^{2} \)
89 \( 1 - 2.42e10T + 2.77e21T^{2} \)
97 \( 1 - 3.60e10T + 7.15e21T^{2} \)
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

−10.39340010708831607510350034951, −9.520139864970083813799986829329, −8.763549612561781711276533184252, −7.60900009343842515041159922644, −7.30723899401759976371977611685, −5.30913511259395115674264508257, −4.33240889102086914786812269309, −2.83764870709683681243653969659, −1.87612068897408924183958315790, −0.17417578859974240725981666721, 0.17417578859974240725981666721, 1.87612068897408924183958315790, 2.83764870709683681243653969659, 4.33240889102086914786812269309, 5.30913511259395115674264508257, 7.30723899401759976371977611685, 7.60900009343842515041159922644, 8.763549612561781711276533184252, 9.520139864970083813799986829329, 10.39340010708831607510350034951

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