Properties

Label 2-177-1.1-c11-0-19
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 $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 60.8·2-s − 243·3-s + 1.65e3·4-s − 4.96e3·5-s + 1.47e4·6-s − 8.70e4·7-s + 2.36e4·8-s + 5.90e4·9-s + 3.02e5·10-s − 6.83e5·11-s − 4.03e5·12-s − 9.74e5·13-s + 5.30e6·14-s + 1.20e6·15-s − 4.83e6·16-s − 6.81e6·17-s − 3.59e6·18-s + 1.81e7·19-s − 8.23e6·20-s + 2.11e7·21-s + 4.15e7·22-s − 3.49e7·23-s − 5.74e6·24-s − 2.42e7·25-s + 5.93e7·26-s − 1.43e7·27-s − 1.44e8·28-s + ⋯
L(s)  = 1  − 1.34·2-s − 0.577·3-s + 0.810·4-s − 0.709·5-s + 0.776·6-s − 1.95·7-s + 0.255·8-s + 0.333·9-s + 0.955·10-s − 1.27·11-s − 0.467·12-s − 0.727·13-s + 2.63·14-s + 0.409·15-s − 1.15·16-s − 1.16·17-s − 0.448·18-s + 1.67·19-s − 0.575·20-s + 1.13·21-s + 1.72·22-s − 1.13·23-s − 0.147·24-s − 0.495·25-s + 0.979·26-s − 0.192·27-s − 1.58·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 60.8T + 2.04e3T^{2} \)
5 \( 1 + 4.96e3T + 4.88e7T^{2} \)
7 \( 1 + 8.70e4T + 1.97e9T^{2} \)
11 \( 1 + 6.83e5T + 2.85e11T^{2} \)
13 \( 1 + 9.74e5T + 1.79e12T^{2} \)
17 \( 1 + 6.81e6T + 3.42e13T^{2} \)
19 \( 1 - 1.81e7T + 1.16e14T^{2} \)
23 \( 1 + 3.49e7T + 9.52e14T^{2} \)
29 \( 1 - 9.30e7T + 1.22e16T^{2} \)
31 \( 1 + 1.05e8T + 2.54e16T^{2} \)
37 \( 1 + 2.32e8T + 1.77e17T^{2} \)
41 \( 1 - 1.02e9T + 5.50e17T^{2} \)
43 \( 1 + 1.24e9T + 9.29e17T^{2} \)
47 \( 1 + 2.18e9T + 2.47e18T^{2} \)
53 \( 1 + 4.40e9T + 9.26e18T^{2} \)
61 \( 1 - 8.44e9T + 4.35e19T^{2} \)
67 \( 1 - 2.45e9T + 1.22e20T^{2} \)
71 \( 1 - 1.56e10T + 2.31e20T^{2} \)
73 \( 1 - 1.89e10T + 3.13e20T^{2} \)
79 \( 1 - 1.99e10T + 7.47e20T^{2} \)
83 \( 1 - 6.31e10T + 1.28e21T^{2} \)
89 \( 1 - 1.84e10T + 2.77e21T^{2} \)
97 \( 1 - 4.38e10T + 7.15e21T^{2} \)
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

−9.875219481800305076401652593780, −9.524537667122289270813623398167, −8.105738556299083999460822267977, −7.29394201018651160108354860232, −6.41621605500750235254369526197, −5.00788506799220357320333916247, −3.54825084968552247214876009451, −2.30990786195103037940662455571, −0.53629755298077713854372451436, 0, 0.53629755298077713854372451436, 2.30990786195103037940662455571, 3.54825084968552247214876009451, 5.00788506799220357320333916247, 6.41621605500750235254369526197, 7.29394201018651160108354860232, 8.105738556299083999460822267977, 9.524537667122289270813623398167, 9.875219481800305076401652593780

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