Properties

Label 2-177-1.1-c11-0-26
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  − 12.4·2-s + 243·3-s − 1.89e3·4-s − 2.62e3·5-s − 3.01e3·6-s + 2.32e4·7-s + 4.89e4·8-s + 5.90e4·9-s + 3.26e4·10-s − 2.19e5·11-s − 4.60e5·12-s + 1.40e6·13-s − 2.89e5·14-s − 6.38e5·15-s + 3.27e6·16-s − 5.70e6·17-s − 7.32e5·18-s + 8.20e5·19-s + 4.98e6·20-s + 5.66e6·21-s + 2.72e6·22-s + 2.97e7·23-s + 1.18e7·24-s − 4.19e7·25-s − 1.74e7·26-s + 1.43e7·27-s − 4.41e7·28-s + ⋯
L(s)  = 1  − 0.274·2-s + 0.577·3-s − 0.924·4-s − 0.376·5-s − 0.158·6-s + 0.523·7-s + 0.527·8-s + 0.333·9-s + 0.103·10-s − 0.411·11-s − 0.533·12-s + 1.04·13-s − 0.143·14-s − 0.217·15-s + 0.780·16-s − 0.974·17-s − 0.0914·18-s + 0.0760·19-s + 0.347·20-s + 0.302·21-s + 0.112·22-s + 0.962·23-s + 0.304·24-s − 0.858·25-s − 0.287·26-s + 0.192·27-s − 0.484·28-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

\(L(6)\) \(\approx\) \(1.707880814\)
\(L(\frac12)\) \(\approx\) \(1.707880814\)
\(L(\frac{13}{2})\) 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 \( 1 + 12.4T + 2.04e3T^{2} \)
5 \( 1 + 2.62e3T + 4.88e7T^{2} \)
7 \( 1 - 2.32e4T + 1.97e9T^{2} \)
11 \( 1 + 2.19e5T + 2.85e11T^{2} \)
13 \( 1 - 1.40e6T + 1.79e12T^{2} \)
17 \( 1 + 5.70e6T + 3.42e13T^{2} \)
19 \( 1 - 8.20e5T + 1.16e14T^{2} \)
23 \( 1 - 2.97e7T + 9.52e14T^{2} \)
29 \( 1 - 9.68e7T + 1.22e16T^{2} \)
31 \( 1 - 6.91e7T + 2.54e16T^{2} \)
37 \( 1 + 4.51e8T + 1.77e17T^{2} \)
41 \( 1 - 4.49e8T + 5.50e17T^{2} \)
43 \( 1 + 1.79e9T + 9.29e17T^{2} \)
47 \( 1 - 5.98e8T + 2.47e18T^{2} \)
53 \( 1 + 3.80e9T + 9.26e18T^{2} \)
61 \( 1 - 9.73e9T + 4.35e19T^{2} \)
67 \( 1 - 2.05e10T + 1.22e20T^{2} \)
71 \( 1 + 1.31e10T + 2.31e20T^{2} \)
73 \( 1 - 2.61e10T + 3.13e20T^{2} \)
79 \( 1 + 3.90e10T + 7.47e20T^{2} \)
83 \( 1 - 2.57e10T + 1.28e21T^{2} \)
89 \( 1 - 4.36e10T + 2.77e21T^{2} \)
97 \( 1 + 4.40e10T + 7.15e21T^{2} \)
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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

−10.55283119125955674515688689764, −9.487031248300338249922030464573, −8.477051007274340154572979473896, −8.101127084255837618514885712681, −6.75823940623821054270699288691, −5.20047226137305383105633965134, −4.27750268280426984226345930034, −3.27307251177710472629984790110, −1.76361458550665314378023377993, −0.62197925963046430875363013529, 0.62197925963046430875363013529, 1.76361458550665314378023377993, 3.27307251177710472629984790110, 4.27750268280426984226345930034, 5.20047226137305383105633965134, 6.75823940623821054270699288691, 8.101127084255837618514885712681, 8.477051007274340154572979473896, 9.487031248300338249922030464573, 10.55283119125955674515688689764

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