Properties

Label 2-177-1.1-c13-0-72
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 63.8·2-s − 729·3-s − 4.11e3·4-s + 1.84e4·5-s + 4.65e4·6-s + 1.20e5·7-s + 7.85e5·8-s + 5.31e5·9-s − 1.17e6·10-s − 3.91e6·11-s + 2.99e6·12-s − 9.55e6·13-s − 7.67e6·14-s − 1.34e7·15-s − 1.64e7·16-s + 4.69e6·17-s − 3.39e7·18-s + 3.61e8·19-s − 7.57e7·20-s − 8.76e7·21-s + 2.50e8·22-s + 1.17e9·23-s − 5.72e8·24-s − 8.81e8·25-s + 6.10e8·26-s − 3.87e8·27-s − 4.94e8·28-s + ⋯
L(s)  = 1  − 0.705·2-s − 0.577·3-s − 0.502·4-s + 0.527·5-s + 0.407·6-s + 0.386·7-s + 1.05·8-s + 0.333·9-s − 0.371·10-s − 0.666·11-s + 0.289·12-s − 0.549·13-s − 0.272·14-s − 0.304·15-s − 0.245·16-s + 0.0471·17-s − 0.235·18-s + 1.76·19-s − 0.264·20-s − 0.223·21-s + 0.470·22-s + 1.65·23-s − 0.611·24-s − 0.722·25-s + 0.387·26-s − 0.192·27-s − 0.194·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 + 63.8T + 8.19e3T^{2} \)
5 \( 1 - 1.84e4T + 1.22e9T^{2} \)
7 \( 1 - 1.20e5T + 9.68e10T^{2} \)
11 \( 1 + 3.91e6T + 3.45e13T^{2} \)
13 \( 1 + 9.55e6T + 3.02e14T^{2} \)
17 \( 1 - 4.69e6T + 9.90e15T^{2} \)
19 \( 1 - 3.61e8T + 4.20e16T^{2} \)
23 \( 1 - 1.17e9T + 5.04e17T^{2} \)
29 \( 1 + 4.59e9T + 1.02e19T^{2} \)
31 \( 1 + 5.94e9T + 2.44e19T^{2} \)
37 \( 1 + 1.48e10T + 2.43e20T^{2} \)
41 \( 1 + 1.83e10T + 9.25e20T^{2} \)
43 \( 1 - 1.62e10T + 1.71e21T^{2} \)
47 \( 1 - 1.36e9T + 5.46e21T^{2} \)
53 \( 1 - 1.60e11T + 2.60e22T^{2} \)
61 \( 1 - 2.01e11T + 1.61e23T^{2} \)
67 \( 1 - 5.67e11T + 5.48e23T^{2} \)
71 \( 1 + 2.83e11T + 1.16e24T^{2} \)
73 \( 1 - 2.34e12T + 1.67e24T^{2} \)
79 \( 1 - 1.78e12T + 4.66e24T^{2} \)
83 \( 1 - 1.42e12T + 8.87e24T^{2} \)
89 \( 1 + 5.50e12T + 2.19e25T^{2} \)
97 \( 1 - 1.04e13T + 6.73e25T^{2} \)
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

−9.731320519171049089550454704620, −9.111454034292048663702990681653, −7.79180783003846359546982720472, −7.11767683243623640279317523812, −5.36207866151048754057034651484, −5.12152979108983607241598094021, −3.56336152024260822717476365267, −1.98601335856450508192677415578, −1.00130808256881413536482051954, 0, 1.00130808256881413536482051954, 1.98601335856450508192677415578, 3.56336152024260822717476365267, 5.12152979108983607241598094021, 5.36207866151048754057034651484, 7.11767683243623640279317523812, 7.79180783003846359546982720472, 9.111454034292048663702990681653, 9.731320519171049089550454704620

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