Properties

Label 2-177-1.1-c7-0-10
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 15.1·2-s − 27·3-s + 102.·4-s − 418.·5-s − 410.·6-s − 814.·7-s − 384.·8-s + 729·9-s − 6.36e3·10-s − 2.42e3·11-s − 2.77e3·12-s + 1.06e4·13-s − 1.23e4·14-s + 1.13e4·15-s − 1.89e4·16-s − 2.67e4·17-s + 1.10e4·18-s + 3.38e4·19-s − 4.30e4·20-s + 2.19e4·21-s − 3.68e4·22-s + 1.50e4·23-s + 1.03e4·24-s + 9.72e4·25-s + 1.62e5·26-s − 1.96e4·27-s − 8.36e4·28-s + ⋯
L(s)  = 1  + 1.34·2-s − 0.577·3-s + 0.802·4-s − 1.49·5-s − 0.775·6-s − 0.897·7-s − 0.265·8-s + 0.333·9-s − 2.01·10-s − 0.548·11-s − 0.463·12-s + 1.34·13-s − 1.20·14-s + 0.865·15-s − 1.15·16-s − 1.31·17-s + 0.447·18-s + 1.13·19-s − 1.20·20-s + 0.518·21-s − 0.736·22-s + 0.258·23-s + 0.153·24-s + 1.24·25-s + 1.80·26-s − 0.192·27-s − 0.720·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.562179637\)
\(L(\frac12)\) \(\approx\) \(1.562179637\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27T \)
59 \( 1 + 2.05e5T \)
good2 \( 1 - 15.1T + 128T^{2} \)
5 \( 1 + 418.T + 7.81e4T^{2} \)
7 \( 1 + 814.T + 8.23e5T^{2} \)
11 \( 1 + 2.42e3T + 1.94e7T^{2} \)
13 \( 1 - 1.06e4T + 6.27e7T^{2} \)
17 \( 1 + 2.67e4T + 4.10e8T^{2} \)
19 \( 1 - 3.38e4T + 8.93e8T^{2} \)
23 \( 1 - 1.50e4T + 3.40e9T^{2} \)
29 \( 1 - 3.25e4T + 1.72e10T^{2} \)
31 \( 1 - 2.64e4T + 2.75e10T^{2} \)
37 \( 1 - 4.97e5T + 9.49e10T^{2} \)
41 \( 1 - 3.66e5T + 1.94e11T^{2} \)
43 \( 1 - 4.93e5T + 2.71e11T^{2} \)
47 \( 1 + 8.25e5T + 5.06e11T^{2} \)
53 \( 1 + 1.78e6T + 1.17e12T^{2} \)
61 \( 1 - 6.28e5T + 3.14e12T^{2} \)
67 \( 1 - 1.85e6T + 6.06e12T^{2} \)
71 \( 1 - 3.64e6T + 9.09e12T^{2} \)
73 \( 1 - 5.50e6T + 1.10e13T^{2} \)
79 \( 1 + 6.04e6T + 1.92e13T^{2} \)
83 \( 1 + 4.72e6T + 2.71e13T^{2} \)
89 \( 1 + 8.75e5T + 4.42e13T^{2} \)
97 \( 1 + 1.26e7T + 8.07e13T^{2} \)
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

−11.35730138710645911532868996242, −11.17035320737670773885456064154, −9.419902525150863547995213401850, −8.104453669501904063470686230707, −6.83946577426399018460218513726, −5.99265139394057209581180233299, −4.71244823466170547480746443763, −3.85977081572376217253325217331, −2.96549384676386045440357783516, −0.55822223480724481547935235938, 0.55822223480724481547935235938, 2.96549384676386045440357783516, 3.85977081572376217253325217331, 4.71244823466170547480746443763, 5.99265139394057209581180233299, 6.83946577426399018460218513726, 8.104453669501904063470686230707, 9.419902525150863547995213401850, 11.17035320737670773885456064154, 11.35730138710645911532868996242

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