Properties

Label 2-177-1.1-c9-0-31
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 9.57·2-s + 81·3-s − 420.·4-s + 279.·5-s − 775.·6-s + 7.95e3·7-s + 8.92e3·8-s + 6.56e3·9-s − 2.68e3·10-s − 2.44e4·11-s − 3.40e4·12-s − 4.18e3·13-s − 7.61e4·14-s + 2.26e4·15-s + 1.29e5·16-s + 1.64e5·17-s − 6.28e4·18-s + 8.26e5·19-s − 1.17e5·20-s + 6.44e5·21-s + 2.34e5·22-s + 8.40e5·23-s + 7.23e5·24-s − 1.87e6·25-s + 4.00e4·26-s + 5.31e5·27-s − 3.34e6·28-s + ⋯
L(s)  = 1  − 0.423·2-s + 0.577·3-s − 0.820·4-s + 0.200·5-s − 0.244·6-s + 1.25·7-s + 0.770·8-s + 0.333·9-s − 0.0848·10-s − 0.503·11-s − 0.473·12-s − 0.0406·13-s − 0.529·14-s + 0.115·15-s + 0.494·16-s + 0.476·17-s − 0.141·18-s + 1.45·19-s − 0.164·20-s + 0.722·21-s + 0.213·22-s + 0.626·23-s + 0.445·24-s − 0.959·25-s + 0.0171·26-s + 0.192·27-s − 1.02·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(2.286075634\)
\(L(\frac12)\) \(\approx\) \(2.286075634\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 + 9.57T + 512T^{2} \)
5 \( 1 - 279.T + 1.95e6T^{2} \)
7 \( 1 - 7.95e3T + 4.03e7T^{2} \)
11 \( 1 + 2.44e4T + 2.35e9T^{2} \)
13 \( 1 + 4.18e3T + 1.06e10T^{2} \)
17 \( 1 - 1.64e5T + 1.18e11T^{2} \)
19 \( 1 - 8.26e5T + 3.22e11T^{2} \)
23 \( 1 - 8.40e5T + 1.80e12T^{2} \)
29 \( 1 - 6.86e6T + 1.45e13T^{2} \)
31 \( 1 + 1.00e7T + 2.64e13T^{2} \)
37 \( 1 - 2.02e6T + 1.29e14T^{2} \)
41 \( 1 + 2.68e7T + 3.27e14T^{2} \)
43 \( 1 - 3.41e7T + 5.02e14T^{2} \)
47 \( 1 + 1.02e7T + 1.11e15T^{2} \)
53 \( 1 + 1.18e7T + 3.29e15T^{2} \)
61 \( 1 - 2.71e7T + 1.16e16T^{2} \)
67 \( 1 + 5.94e7T + 2.72e16T^{2} \)
71 \( 1 + 8.70e7T + 4.58e16T^{2} \)
73 \( 1 - 5.93e7T + 5.88e16T^{2} \)
79 \( 1 - 4.90e8T + 1.19e17T^{2} \)
83 \( 1 + 8.27e7T + 1.86e17T^{2} \)
89 \( 1 - 6.23e8T + 3.50e17T^{2} \)
97 \( 1 - 1.98e8T + 7.60e17T^{2} \)
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

−10.78165569487064066043565121566, −9.842787217029827774225849539335, −8.959335310608453211432820180443, −8.033388708751372268778604900693, −7.41792386188500815230512127539, −5.45399326176866412246713191055, −4.68364570489734837345435723362, −3.35720624583824840527227146028, −1.83103933767744685838095614938, −0.826592806917573630190238037028, 0.826592806917573630190238037028, 1.83103933767744685838095614938, 3.35720624583824840527227146028, 4.68364570489734837345435723362, 5.45399326176866412246713191055, 7.41792386188500815230512127539, 8.033388708751372268778604900693, 8.959335310608453211432820180443, 9.842787217029827774225849539335, 10.78165569487064066043565121566

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