Properties

Label 2-177-1.1-c9-0-83
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  + 30.4·2-s + 81·3-s + 418.·4-s + 1.05e3·5-s + 2.47e3·6-s − 5.89e3·7-s − 2.86e3·8-s + 6.56e3·9-s + 3.20e4·10-s + 2.63e4·11-s + 3.38e4·12-s − 7.11e4·13-s − 1.79e5·14-s + 8.51e4·15-s − 3.01e5·16-s − 6.60e5·17-s + 2.00e5·18-s − 7.13e5·19-s + 4.39e5·20-s − 4.77e5·21-s + 8.04e5·22-s + 1.72e6·23-s − 2.31e5·24-s − 8.48e5·25-s − 2.17e6·26-s + 5.31e5·27-s − 2.46e6·28-s + ⋯
L(s)  = 1  + 1.34·2-s + 0.577·3-s + 0.816·4-s + 0.751·5-s + 0.778·6-s − 0.927·7-s − 0.247·8-s + 0.333·9-s + 1.01·10-s + 0.543·11-s + 0.471·12-s − 0.691·13-s − 1.24·14-s + 0.434·15-s − 1.14·16-s − 1.91·17-s + 0.449·18-s − 1.25·19-s + 0.614·20-s − 0.535·21-s + 0.732·22-s + 1.28·23-s − 0.142·24-s − 0.434·25-s − 0.931·26-s + 0.192·27-s − 0.757·28-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 \( 1 - 30.4T + 512T^{2} \)
5 \( 1 - 1.05e3T + 1.95e6T^{2} \)
7 \( 1 + 5.89e3T + 4.03e7T^{2} \)
11 \( 1 - 2.63e4T + 2.35e9T^{2} \)
13 \( 1 + 7.11e4T + 1.06e10T^{2} \)
17 \( 1 + 6.60e5T + 1.18e11T^{2} \)
19 \( 1 + 7.13e5T + 3.22e11T^{2} \)
23 \( 1 - 1.72e6T + 1.80e12T^{2} \)
29 \( 1 + 8.82e5T + 1.45e13T^{2} \)
31 \( 1 - 8.02e6T + 2.64e13T^{2} \)
37 \( 1 + 1.38e7T + 1.29e14T^{2} \)
41 \( 1 + 1.60e7T + 3.27e14T^{2} \)
43 \( 1 + 1.63e7T + 5.02e14T^{2} \)
47 \( 1 - 4.25e7T + 1.11e15T^{2} \)
53 \( 1 - 5.47e6T + 3.29e15T^{2} \)
61 \( 1 + 1.55e8T + 1.16e16T^{2} \)
67 \( 1 - 1.19e8T + 2.72e16T^{2} \)
71 \( 1 - 5.29e7T + 4.58e16T^{2} \)
73 \( 1 + 1.44e8T + 5.88e16T^{2} \)
79 \( 1 - 3.12e8T + 1.19e17T^{2} \)
83 \( 1 - 5.07e7T + 1.86e17T^{2} \)
89 \( 1 - 6.65e8T + 3.50e17T^{2} \)
97 \( 1 + 1.11e9T + 7.60e17T^{2} \)
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

−10.57010821814302143060062219028, −9.405261459674816257531984613683, −8.737423433217838931212422129051, −6.78273648662214770460204681592, −6.38137841564374655769851064108, −4.97283576602514331978895898898, −4.04139646842488964683699978615, −2.87579503202539533591865381704, −2.04047201753696365412129847984, 0, 2.04047201753696365412129847984, 2.87579503202539533591865381704, 4.04139646842488964683699978615, 4.97283576602514331978895898898, 6.38137841564374655769851064108, 6.78273648662214770460204681592, 8.737423433217838931212422129051, 9.405261459674816257531984613683, 10.57010821814302143060062219028

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