Properties

Label 2-177-1.1-c11-0-81
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  − 19.9·2-s + 243·3-s − 1.65e3·4-s + 1.11e4·5-s − 4.83e3·6-s − 2.76e4·7-s + 7.36e4·8-s + 5.90e4·9-s − 2.22e5·10-s − 1.04e6·11-s − 4.01e5·12-s + 1.62e5·13-s + 5.50e5·14-s + 2.71e6·15-s + 1.91e6·16-s + 6.41e6·17-s − 1.17e6·18-s + 1.86e7·19-s − 1.84e7·20-s − 6.72e6·21-s + 2.07e7·22-s − 5.10e7·23-s + 1.78e7·24-s + 7.59e7·25-s − 3.23e6·26-s + 1.43e7·27-s + 4.56e7·28-s + ⋯
L(s)  = 1  − 0.439·2-s + 0.577·3-s − 0.806·4-s + 1.59·5-s − 0.253·6-s − 0.622·7-s + 0.794·8-s + 0.333·9-s − 0.702·10-s − 1.94·11-s − 0.465·12-s + 0.121·13-s + 0.273·14-s + 0.922·15-s + 0.457·16-s + 1.09·17-s − 0.146·18-s + 1.72·19-s − 1.28·20-s − 0.359·21-s + 0.857·22-s − 1.65·23-s + 0.458·24-s + 1.55·25-s − 0.0533·26-s + 0.192·27-s + 0.501·28-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 \( 1 + 19.9T + 2.04e3T^{2} \)
5 \( 1 - 1.11e4T + 4.88e7T^{2} \)
7 \( 1 + 2.76e4T + 1.97e9T^{2} \)
11 \( 1 + 1.04e6T + 2.85e11T^{2} \)
13 \( 1 - 1.62e5T + 1.79e12T^{2} \)
17 \( 1 - 6.41e6T + 3.42e13T^{2} \)
19 \( 1 - 1.86e7T + 1.16e14T^{2} \)
23 \( 1 + 5.10e7T + 9.52e14T^{2} \)
29 \( 1 + 2.37e6T + 1.22e16T^{2} \)
31 \( 1 + 2.59e8T + 2.54e16T^{2} \)
37 \( 1 - 1.23e8T + 1.77e17T^{2} \)
41 \( 1 - 1.74e8T + 5.50e17T^{2} \)
43 \( 1 + 1.00e9T + 9.29e17T^{2} \)
47 \( 1 - 1.25e9T + 2.47e18T^{2} \)
53 \( 1 - 1.52e9T + 9.26e18T^{2} \)
61 \( 1 - 6.43e9T + 4.35e19T^{2} \)
67 \( 1 + 1.91e9T + 1.22e20T^{2} \)
71 \( 1 + 1.22e10T + 2.31e20T^{2} \)
73 \( 1 - 2.33e10T + 3.13e20T^{2} \)
79 \( 1 + 1.85e10T + 7.47e20T^{2} \)
83 \( 1 - 3.10e10T + 1.28e21T^{2} \)
89 \( 1 + 1.78e10T + 2.77e21T^{2} \)
97 \( 1 + 1.24e11T + 7.15e21T^{2} \)
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

−9.822086349353463960620118841767, −9.527835392729414172642484879038, −8.235814912403962539983002712005, −7.42144066691172764721042504061, −5.70330610751188580892641653777, −5.23690711978849626564780643663, −3.49979778754165094793765083720, −2.42975265495549634949390110587, −1.31787059532216526849463281224, 0, 1.31787059532216526849463281224, 2.42975265495549634949390110587, 3.49979778754165094793765083720, 5.23690711978849626564780643663, 5.70330610751188580892641653777, 7.42144066691172764721042504061, 8.235814912403962539983002712005, 9.527835392729414172642484879038, 9.822086349353463960620118841767

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