Properties

Label 2-177-1.1-c9-0-2
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  − 38.0·2-s + 81·3-s + 939.·4-s − 310.·5-s − 3.08e3·6-s − 4.49e3·7-s − 1.62e4·8-s + 6.56e3·9-s + 1.18e4·10-s − 5.77e4·11-s + 7.60e4·12-s − 1.58e5·13-s + 1.71e5·14-s − 2.51e4·15-s + 1.39e5·16-s − 4.06e5·17-s − 2.49e5·18-s − 5.88e5·19-s − 2.91e5·20-s − 3.63e5·21-s + 2.20e6·22-s − 2.03e6·23-s − 1.31e6·24-s − 1.85e6·25-s + 6.02e6·26-s + 5.31e5·27-s − 4.21e6·28-s + ⋯
L(s)  = 1  − 1.68·2-s + 0.577·3-s + 1.83·4-s − 0.221·5-s − 0.971·6-s − 0.707·7-s − 1.40·8-s + 0.333·9-s + 0.373·10-s − 1.18·11-s + 1.05·12-s − 1.53·13-s + 1.19·14-s − 0.128·15-s + 0.530·16-s − 1.18·17-s − 0.561·18-s − 1.03·19-s − 0.407·20-s − 0.408·21-s + 2.00·22-s − 1.51·23-s − 0.810·24-s − 0.950·25-s + 2.58·26-s + 0.192·27-s − 1.29·28-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

\(L(5)\) \(\approx\) \(0.09298824051\)
\(L(\frac12)\) \(\approx\) \(0.09298824051\)
\(L(\frac{11}{2})\) 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 \( 1 + 38.0T + 512T^{2} \)
5 \( 1 + 310.T + 1.95e6T^{2} \)
7 \( 1 + 4.49e3T + 4.03e7T^{2} \)
11 \( 1 + 5.77e4T + 2.35e9T^{2} \)
13 \( 1 + 1.58e5T + 1.06e10T^{2} \)
17 \( 1 + 4.06e5T + 1.18e11T^{2} \)
19 \( 1 + 5.88e5T + 3.22e11T^{2} \)
23 \( 1 + 2.03e6T + 1.80e12T^{2} \)
29 \( 1 - 4.76e6T + 1.45e13T^{2} \)
31 \( 1 - 1.42e6T + 2.64e13T^{2} \)
37 \( 1 + 4.04e5T + 1.29e14T^{2} \)
41 \( 1 + 7.85e6T + 3.27e14T^{2} \)
43 \( 1 - 6.47e6T + 5.02e14T^{2} \)
47 \( 1 - 1.12e7T + 1.11e15T^{2} \)
53 \( 1 - 5.38e7T + 3.29e15T^{2} \)
61 \( 1 + 1.25e8T + 1.16e16T^{2} \)
67 \( 1 + 8.34e7T + 2.72e16T^{2} \)
71 \( 1 + 2.26e8T + 4.58e16T^{2} \)
73 \( 1 - 3.98e7T + 5.88e16T^{2} \)
79 \( 1 + 4.27e8T + 1.19e17T^{2} \)
83 \( 1 - 6.54e8T + 1.86e17T^{2} \)
89 \( 1 + 8.81e8T + 3.50e17T^{2} \)
97 \( 1 + 1.21e8T + 7.60e17T^{2} \)
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

−10.35521623430501592898865174908, −10.04941410590721485889094886486, −8.974867262487331109492901740266, −8.097522901530690439223590727263, −7.37239664184529203220695007751, −6.32042722582928489484429184713, −4.46642572932949885282914351365, −2.68543876446031308524211029722, −2.04783957312966108409223313617, −0.17282345261579597758141864504, 0.17282345261579597758141864504, 2.04783957312966108409223313617, 2.68543876446031308524211029722, 4.46642572932949885282914351365, 6.32042722582928489484429184713, 7.37239664184529203220695007751, 8.097522901530690439223590727263, 8.974867262487331109492901740266, 10.04941410590721485889094886486, 10.35521623430501592898865174908

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