Properties

Label 2-177-1.1-c11-0-80
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 89.5·2-s − 243·3-s + 5.97e3·4-s − 2.39e3·5-s − 2.17e4·6-s + 8.27e4·7-s + 3.51e5·8-s + 5.90e4·9-s − 2.14e5·10-s + 7.79e5·11-s − 1.45e6·12-s − 3.67e5·13-s + 7.41e6·14-s + 5.81e5·15-s + 1.92e7·16-s + 3.35e6·17-s + 5.28e6·18-s − 1.92e6·19-s − 1.43e7·20-s − 2.01e7·21-s + 6.98e7·22-s − 3.41e7·23-s − 8.54e7·24-s − 4.30e7·25-s − 3.29e7·26-s − 1.43e7·27-s + 4.94e8·28-s + ⋯
L(s)  = 1  + 1.97·2-s − 0.577·3-s + 2.91·4-s − 0.342·5-s − 1.14·6-s + 1.86·7-s + 3.79·8-s + 0.333·9-s − 0.678·10-s + 1.45·11-s − 1.68·12-s − 0.274·13-s + 3.68·14-s + 0.197·15-s + 4.59·16-s + 0.573·17-s + 0.659·18-s − 0.178·19-s − 0.999·20-s − 1.07·21-s + 2.88·22-s − 1.10·23-s − 2.19·24-s − 0.882·25-s − 0.543·26-s − 0.192·27-s + 5.43·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(10.44013098\)
\(L(\frac12)\) \(\approx\) \(10.44013098\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 + 7.14e8T \)
good2 \( 1 - 89.5T + 2.04e3T^{2} \)
5 \( 1 + 2.39e3T + 4.88e7T^{2} \)
7 \( 1 - 8.27e4T + 1.97e9T^{2} \)
11 \( 1 - 7.79e5T + 2.85e11T^{2} \)
13 \( 1 + 3.67e5T + 1.79e12T^{2} \)
17 \( 1 - 3.35e6T + 3.42e13T^{2} \)
19 \( 1 + 1.92e6T + 1.16e14T^{2} \)
23 \( 1 + 3.41e7T + 9.52e14T^{2} \)
29 \( 1 + 2.65e7T + 1.22e16T^{2} \)
31 \( 1 - 1.71e8T + 2.54e16T^{2} \)
37 \( 1 + 7.07e8T + 1.77e17T^{2} \)
41 \( 1 + 2.76e8T + 5.50e17T^{2} \)
43 \( 1 + 9.77e8T + 9.29e17T^{2} \)
47 \( 1 - 2.45e9T + 2.47e18T^{2} \)
53 \( 1 + 3.75e9T + 9.26e18T^{2} \)
61 \( 1 - 1.05e9T + 4.35e19T^{2} \)
67 \( 1 + 4.59e9T + 1.22e20T^{2} \)
71 \( 1 - 1.72e10T + 2.31e20T^{2} \)
73 \( 1 - 2.89e9T + 3.13e20T^{2} \)
79 \( 1 + 5.08e10T + 7.47e20T^{2} \)
83 \( 1 - 3.23e10T + 1.28e21T^{2} \)
89 \( 1 - 1.57e10T + 2.77e21T^{2} \)
97 \( 1 - 1.26e11T + 7.15e21T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.30088083206591501250457118190, −10.29690066619403446614261888717, −8.145569371057700546869829686759, −7.23056595479817638434278616280, −6.16783470192277308214757963834, −5.21882102297128147312225518168, −4.41355136098144941503621250830, −3.69903970037825725108062669318, −2.00240563608223030868577907964, −1.31289105435087891076286762035, 1.31289105435087891076286762035, 2.00240563608223030868577907964, 3.69903970037825725108062669318, 4.41355136098144941503621250830, 5.21882102297128147312225518168, 6.16783470192277308214757963834, 7.23056595479817638434278616280, 8.145569371057700546869829686759, 10.29690066619403446614261888717, 11.30088083206591501250457118190

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