Properties

Label 2-177-1.1-c9-0-73
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 $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 34.8·2-s − 81·3-s + 700.·4-s − 913.·5-s − 2.82e3·6-s − 1.30e3·7-s + 6.56e3·8-s + 6.56e3·9-s − 3.18e4·10-s + 6.77e4·11-s − 5.67e4·12-s + 3.55e4·13-s − 4.54e4·14-s + 7.39e4·15-s − 1.30e5·16-s + 1.22e5·17-s + 2.28e5·18-s − 4.26e5·19-s − 6.39e5·20-s + 1.05e5·21-s + 2.36e6·22-s + 6.28e4·23-s − 5.31e5·24-s − 1.11e6·25-s + 1.23e6·26-s − 5.31e5·27-s − 9.14e5·28-s + ⋯
L(s)  = 1  + 1.53·2-s − 0.577·3-s + 1.36·4-s − 0.653·5-s − 0.888·6-s − 0.205·7-s + 0.566·8-s + 0.333·9-s − 1.00·10-s + 1.39·11-s − 0.789·12-s + 0.345·13-s − 0.316·14-s + 0.377·15-s − 0.496·16-s + 0.356·17-s + 0.512·18-s − 0.750·19-s − 0.893·20-s + 0.118·21-s + 2.14·22-s + 0.0467·23-s − 0.326·24-s − 0.572·25-s + 0.531·26-s − 0.192·27-s − 0.281·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 34.8T + 512T^{2} \)
5 \( 1 + 913.T + 1.95e6T^{2} \)
7 \( 1 + 1.30e3T + 4.03e7T^{2} \)
11 \( 1 - 6.77e4T + 2.35e9T^{2} \)
13 \( 1 - 3.55e4T + 1.06e10T^{2} \)
17 \( 1 - 1.22e5T + 1.18e11T^{2} \)
19 \( 1 + 4.26e5T + 3.22e11T^{2} \)
23 \( 1 - 6.28e4T + 1.80e12T^{2} \)
29 \( 1 + 6.02e5T + 1.45e13T^{2} \)
31 \( 1 - 5.48e6T + 2.64e13T^{2} \)
37 \( 1 + 6.71e6T + 1.29e14T^{2} \)
41 \( 1 + 3.26e7T + 3.27e14T^{2} \)
43 \( 1 + 5.20e5T + 5.02e14T^{2} \)
47 \( 1 + 4.65e7T + 1.11e15T^{2} \)
53 \( 1 + 9.11e7T + 3.29e15T^{2} \)
61 \( 1 - 6.62e7T + 1.16e16T^{2} \)
67 \( 1 + 1.85e7T + 2.72e16T^{2} \)
71 \( 1 - 3.02e8T + 4.58e16T^{2} \)
73 \( 1 + 2.05e8T + 5.88e16T^{2} \)
79 \( 1 - 4.56e8T + 1.19e17T^{2} \)
83 \( 1 + 3.38e8T + 1.86e17T^{2} \)
89 \( 1 + 3.61e8T + 3.50e17T^{2} \)
97 \( 1 + 5.78e8T + 7.60e17T^{2} \)
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

−11.09965979013041773981301494848, −9.695652979837882563732441304005, −8.306992720342373937995428053360, −6.77756883450743897541084608180, −6.25569716946456254709192508444, −5.00490612404437442767401272590, −4.06166750704303476630442869250, −3.31306124249898855011009179449, −1.60734881256800641892601713745, 0, 1.60734881256800641892601713745, 3.31306124249898855011009179449, 4.06166750704303476630442869250, 5.00490612404437442767401272590, 6.25569716946456254709192508444, 6.77756883450743897541084608180, 8.306992720342373937995428053360, 9.695652979837882563732441304005, 11.09965979013041773981301494848

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