Properties

Label 2-177-1.1-c13-0-105
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 152.·2-s + 729·3-s + 1.51e4·4-s − 3.72e3·5-s + 1.11e5·6-s + 5.89e5·7-s + 1.05e6·8-s + 5.31e5·9-s − 5.69e5·10-s + 5.43e6·11-s + 1.10e7·12-s + 2.28e7·13-s + 9.00e7·14-s − 2.71e6·15-s + 3.74e7·16-s − 1.09e7·17-s + 8.11e7·18-s + 2.75e8·19-s − 5.63e7·20-s + 4.29e8·21-s + 8.29e8·22-s − 7.85e7·23-s + 7.70e8·24-s − 1.20e9·25-s + 3.49e9·26-s + 3.87e8·27-s + 8.91e9·28-s + ⋯
L(s)  = 1  + 1.68·2-s + 0.577·3-s + 1.84·4-s − 0.106·5-s + 0.973·6-s + 1.89·7-s + 1.42·8-s + 0.333·9-s − 0.179·10-s + 0.924·11-s + 1.06·12-s + 1.31·13-s + 3.19·14-s − 0.0616·15-s + 0.558·16-s − 0.110·17-s + 0.562·18-s + 1.34·19-s − 0.196·20-s + 1.09·21-s + 1.55·22-s − 0.110·23-s + 0.822·24-s − 0.988·25-s + 2.21·26-s + 0.192·27-s + 3.49·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(12.50534452\)
\(L(\frac12)\) \(\approx\) \(12.50534452\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 - 152.T + 8.19e3T^{2} \)
5 \( 1 + 3.72e3T + 1.22e9T^{2} \)
7 \( 1 - 5.89e5T + 9.68e10T^{2} \)
11 \( 1 - 5.43e6T + 3.45e13T^{2} \)
13 \( 1 - 2.28e7T + 3.02e14T^{2} \)
17 \( 1 + 1.09e7T + 9.90e15T^{2} \)
19 \( 1 - 2.75e8T + 4.20e16T^{2} \)
23 \( 1 + 7.85e7T + 5.04e17T^{2} \)
29 \( 1 + 3.24e8T + 1.02e19T^{2} \)
31 \( 1 + 7.36e9T + 2.44e19T^{2} \)
37 \( 1 + 1.97e10T + 2.43e20T^{2} \)
41 \( 1 - 3.27e10T + 9.25e20T^{2} \)
43 \( 1 - 1.31e10T + 1.71e21T^{2} \)
47 \( 1 + 1.06e11T + 5.46e21T^{2} \)
53 \( 1 + 1.42e11T + 2.60e22T^{2} \)
61 \( 1 + 2.06e11T + 1.61e23T^{2} \)
67 \( 1 - 3.95e11T + 5.48e23T^{2} \)
71 \( 1 - 9.27e11T + 1.16e24T^{2} \)
73 \( 1 + 1.26e12T + 1.67e24T^{2} \)
79 \( 1 - 1.43e12T + 4.66e24T^{2} \)
83 \( 1 - 2.65e12T + 8.87e24T^{2} \)
89 \( 1 + 4.12e12T + 2.19e25T^{2} \)
97 \( 1 + 8.48e12T + 6.73e25T^{2} \)
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

−10.99309024133946449795145074739, −9.202003311031081280068625953529, −8.114625423731051013673483361293, −7.19556070835668151430216127721, −5.90877241834864070188411710015, −5.02534215440454342776804015297, −4.05437164347678688642649353710, −3.41025699563988051452552566809, −1.93807230427128209455787064614, −1.34242202186277239982575705617, 1.34242202186277239982575705617, 1.93807230427128209455787064614, 3.41025699563988051452552566809, 4.05437164347678688642649353710, 5.02534215440454342776804015297, 5.90877241834864070188411710015, 7.19556070835668151430216127721, 8.114625423731051013673483361293, 9.202003311031081280068625953529, 10.99309024133946449795145074739

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