Properties

Label 2-177-1.1-c13-0-32
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  − 59.5·2-s + 729·3-s − 4.64e3·4-s − 4.51e4·5-s − 4.34e4·6-s − 1.34e5·7-s + 7.64e5·8-s + 5.31e5·9-s + 2.69e6·10-s + 9.07e6·11-s − 3.38e6·12-s − 1.72e6·13-s + 8.02e6·14-s − 3.29e7·15-s − 7.48e6·16-s + 5.25e7·17-s − 3.16e7·18-s + 1.91e8·19-s + 2.09e8·20-s − 9.82e7·21-s − 5.40e8·22-s + 6.96e8·23-s + 5.57e8·24-s + 8.21e8·25-s + 1.02e8·26-s + 3.87e8·27-s + 6.26e8·28-s + ⋯
L(s)  = 1  − 0.658·2-s + 0.577·3-s − 0.567·4-s − 1.29·5-s − 0.379·6-s − 0.433·7-s + 1.03·8-s + 0.333·9-s + 0.851·10-s + 1.54·11-s − 0.327·12-s − 0.0990·13-s + 0.285·14-s − 0.746·15-s − 0.111·16-s + 0.528·17-s − 0.219·18-s + 0.933·19-s + 0.733·20-s − 0.250·21-s − 1.01·22-s + 0.980·23-s + 0.595·24-s + 0.672·25-s + 0.0651·26-s + 0.192·27-s + 0.245·28-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

\(L(7)\) \(\approx\) \(1.323414780\)
\(L(\frac12)\) \(\approx\) \(1.323414780\)
\(L(\frac{15}{2})\) 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 \( 1 + 59.5T + 8.19e3T^{2} \)
5 \( 1 + 4.51e4T + 1.22e9T^{2} \)
7 \( 1 + 1.34e5T + 9.68e10T^{2} \)
11 \( 1 - 9.07e6T + 3.45e13T^{2} \)
13 \( 1 + 1.72e6T + 3.02e14T^{2} \)
17 \( 1 - 5.25e7T + 9.90e15T^{2} \)
19 \( 1 - 1.91e8T + 4.20e16T^{2} \)
23 \( 1 - 6.96e8T + 5.04e17T^{2} \)
29 \( 1 + 5.96e8T + 1.02e19T^{2} \)
31 \( 1 - 4.04e9T + 2.44e19T^{2} \)
37 \( 1 - 5.23e9T + 2.43e20T^{2} \)
41 \( 1 + 5.61e10T + 9.25e20T^{2} \)
43 \( 1 - 3.71e10T + 1.71e21T^{2} \)
47 \( 1 + 7.30e10T + 5.46e21T^{2} \)
53 \( 1 + 6.59e10T + 2.60e22T^{2} \)
61 \( 1 - 5.17e11T + 1.61e23T^{2} \)
67 \( 1 - 1.40e11T + 5.48e23T^{2} \)
71 \( 1 - 7.79e11T + 1.16e24T^{2} \)
73 \( 1 + 1.35e12T + 1.67e24T^{2} \)
79 \( 1 - 1.33e12T + 4.66e24T^{2} \)
83 \( 1 + 9.97e11T + 8.87e24T^{2} \)
89 \( 1 - 6.23e12T + 2.19e25T^{2} \)
97 \( 1 - 8.66e12T + 6.73e25T^{2} \)
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

−9.968591896324602268589010151832, −9.261218529006117894380779456678, −8.420423284701108870946033408967, −7.61018345733953683818154783882, −6.70773329453970297592163537819, −4.91040257686962452918709502798, −3.88739928980885473644693625968, −3.24847903982026654814725199111, −1.40021003253958657076769569374, −0.59134895137133716538260907384, 0.59134895137133716538260907384, 1.40021003253958657076769569374, 3.24847903982026654814725199111, 3.88739928980885473644693625968, 4.91040257686962452918709502798, 6.70773329453970297592163537819, 7.61018345733953683818154783882, 8.420423284701108870946033408967, 9.261218529006117894380779456678, 9.968591896324602268589010151832

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