Properties

Label 2-177-1.1-c13-0-24
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 $0$

Origins

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 55.7·2-s + 729·3-s − 5.08e3·4-s + 3.53e4·5-s − 4.06e4·6-s + 1.12e5·7-s + 7.40e5·8-s + 5.31e5·9-s − 1.97e6·10-s − 7.74e6·11-s − 3.70e6·12-s − 2.59e7·13-s − 6.29e6·14-s + 2.57e7·15-s + 3.59e5·16-s − 1.29e8·17-s − 2.96e7·18-s − 3.90e8·19-s − 1.79e8·20-s + 8.22e7·21-s + 4.31e8·22-s − 7.02e8·23-s + 5.39e8·24-s + 3.17e7·25-s + 1.44e9·26-s + 3.87e8·27-s − 5.73e8·28-s + ⋯
L(s)  = 1  − 0.616·2-s + 0.577·3-s − 0.620·4-s + 1.01·5-s − 0.355·6-s + 0.362·7-s + 0.998·8-s + 0.333·9-s − 0.624·10-s − 1.31·11-s − 0.358·12-s − 1.48·13-s − 0.223·14-s + 0.584·15-s + 0.00535·16-s − 1.30·17-s − 0.205·18-s − 1.90·19-s − 0.628·20-s + 0.209·21-s + 0.811·22-s − 0.989·23-s + 0.576·24-s + 0.0260·25-s + 0.917·26-s + 0.192·27-s − 0.224·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(1.038508485\)
\(L(\frac12)\) \(\approx\) \(1.038508485\)
\(L(\frac{15}{2})\) 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 \( 1 + 55.7T + 8.19e3T^{2} \)
5 \( 1 - 3.53e4T + 1.22e9T^{2} \)
7 \( 1 - 1.12e5T + 9.68e10T^{2} \)
11 \( 1 + 7.74e6T + 3.45e13T^{2} \)
13 \( 1 + 2.59e7T + 3.02e14T^{2} \)
17 \( 1 + 1.29e8T + 9.90e15T^{2} \)
19 \( 1 + 3.90e8T + 4.20e16T^{2} \)
23 \( 1 + 7.02e8T + 5.04e17T^{2} \)
29 \( 1 - 4.55e9T + 1.02e19T^{2} \)
31 \( 1 + 7.21e8T + 2.44e19T^{2} \)
37 \( 1 - 1.70e10T + 2.43e20T^{2} \)
41 \( 1 - 1.42e10T + 9.25e20T^{2} \)
43 \( 1 - 8.71e9T + 1.71e21T^{2} \)
47 \( 1 - 6.41e10T + 5.46e21T^{2} \)
53 \( 1 + 2.07e11T + 2.60e22T^{2} \)
61 \( 1 - 7.37e11T + 1.61e23T^{2} \)
67 \( 1 + 9.39e11T + 5.48e23T^{2} \)
71 \( 1 - 7.47e11T + 1.16e24T^{2} \)
73 \( 1 - 2.15e12T + 1.67e24T^{2} \)
79 \( 1 - 2.70e12T + 4.66e24T^{2} \)
83 \( 1 - 7.98e11T + 8.87e24T^{2} \)
89 \( 1 - 1.42e12T + 2.19e25T^{2} \)
97 \( 1 - 5.33e12T + 6.73e25T^{2} \)
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

−10.11441484692250244142859110241, −9.393231959665392437268709876005, −8.397590092548133326772212913013, −7.73491240470450609487319649095, −6.38294240682264867361678105664, −4.99915157163840013151617513029, −4.33425661603714715740835946933, −2.38490635715471489585524733937, −2.07268080368435921651758984144, −0.43722344247090532897748952188, 0.43722344247090532897748952188, 2.07268080368435921651758984144, 2.38490635715471489585524733937, 4.33425661603714715740835946933, 4.99915157163840013151617513029, 6.38294240682264867361678105664, 7.73491240470450609487319649095, 8.397590092548133326772212913013, 9.393231959665392437268709876005, 10.11441484692250244142859110241

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