Properties

Label 2-177-1.1-c11-0-51
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  − 79.2·2-s − 243·3-s + 4.22e3·4-s + 9.67e3·5-s + 1.92e4·6-s + 8.48e4·7-s − 1.72e5·8-s + 5.90e4·9-s − 7.66e5·10-s + 4.65e5·11-s − 1.02e6·12-s + 1.10e6·13-s − 6.72e6·14-s − 2.35e6·15-s + 5.03e6·16-s − 2.02e6·17-s − 4.67e6·18-s − 5.75e6·19-s + 4.09e7·20-s − 2.06e7·21-s − 3.68e7·22-s + 3.68e6·23-s + 4.19e7·24-s + 4.47e7·25-s − 8.72e7·26-s − 1.43e7·27-s + 3.58e8·28-s + ⋯
L(s)  = 1  − 1.75·2-s − 0.577·3-s + 2.06·4-s + 1.38·5-s + 1.01·6-s + 1.90·7-s − 1.86·8-s + 0.333·9-s − 2.42·10-s + 0.871·11-s − 1.19·12-s + 0.822·13-s − 3.34·14-s − 0.799·15-s + 1.19·16-s − 0.346·17-s − 0.583·18-s − 0.533·19-s + 2.85·20-s − 1.10·21-s − 1.52·22-s + 0.119·23-s + 1.07·24-s + 0.915·25-s − 1.44·26-s − 0.192·27-s + 3.94·28-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

\(L(6)\) \(\approx\) \(1.729518497\)
\(L(\frac12)\) \(\approx\) \(1.729518497\)
\(L(\frac{13}{2})\) 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 \( 1 + 79.2T + 2.04e3T^{2} \)
5 \( 1 - 9.67e3T + 4.88e7T^{2} \)
7 \( 1 - 8.48e4T + 1.97e9T^{2} \)
11 \( 1 - 4.65e5T + 2.85e11T^{2} \)
13 \( 1 - 1.10e6T + 1.79e12T^{2} \)
17 \( 1 + 2.02e6T + 3.42e13T^{2} \)
19 \( 1 + 5.75e6T + 1.16e14T^{2} \)
23 \( 1 - 3.68e6T + 9.52e14T^{2} \)
29 \( 1 - 2.53e7T + 1.22e16T^{2} \)
31 \( 1 - 1.59e8T + 2.54e16T^{2} \)
37 \( 1 - 6.00e7T + 1.77e17T^{2} \)
41 \( 1 - 1.17e8T + 5.50e17T^{2} \)
43 \( 1 - 1.58e9T + 9.29e17T^{2} \)
47 \( 1 + 2.01e9T + 2.47e18T^{2} \)
53 \( 1 - 3.31e8T + 9.26e18T^{2} \)
61 \( 1 + 8.69e9T + 4.35e19T^{2} \)
67 \( 1 + 3.59e9T + 1.22e20T^{2} \)
71 \( 1 - 1.77e10T + 2.31e20T^{2} \)
73 \( 1 - 9.43e9T + 3.13e20T^{2} \)
79 \( 1 - 3.35e10T + 7.47e20T^{2} \)
83 \( 1 + 5.21e10T + 1.28e21T^{2} \)
89 \( 1 - 6.06e10T + 2.77e21T^{2} \)
97 \( 1 - 3.05e10T + 7.15e21T^{2} \)
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

−10.68353645912835378230415274600, −9.579174951570397274305249090841, −8.764642188503243841878926571909, −7.944547272475775471029028048891, −6.68367809528780167791974939596, −5.87493511310249448404553272656, −4.56271275279816298393662605213, −2.23536322649072012068437650963, −1.50562472259324936838526154998, −0.913114810268975483875699281031, 0.913114810268975483875699281031, 1.50562472259324936838526154998, 2.23536322649072012068437650963, 4.56271275279816298393662605213, 5.87493511310249448404553272656, 6.68367809528780167791974939596, 7.944547272475775471029028048891, 8.764642188503243841878926571909, 9.579174951570397274305249090841, 10.68353645912835378230415274600

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